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Mathematics LibreTexts

9.5: Matrices and Matrix Operations

Matrices and Matrix Operations
In this section, you will:
  • Find the sum and difference of two matrices.
  • Find scalar multiples of a matrix.
  • Find the product of two matrices.
<figure class="small" id="Figure_09_05_001" style="color: rgb(0, 0, 0); font-family: 'Times New Roman'; font-size: medium; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 1; word-spacing: 0px; -webkit-text-stroke-width: 0px;"> <figcaption>(credit: “SD Dirk,” Flickr)</figcaption> </figure>

Two club soccer teams, the Wildcats and the Mud Cats, are hoping to obtain new equipment for an upcoming season. [link]shows the needs of both teams.

  Wildcats Mud Cats
Goals 6 10
Balls 30 24
Jerseys 14 20

A goal costs $300; a ball costs $10; and a jersey costs $30. How can we find the total cost for the equipment needed for each team? In this section, we discover a method in which the data in the soccer equipment table can be displayed and used for calculating other information. Then, we will be able to calculate the cost of the equipment.

Finding the Sum and Difference of Two Matrices

To solve a problem like the one described for the soccer teams, we can use a matrix, which is a rectangular array of numbers. A row in a matrix is a set of numbers that are aligned horizontally. A column in a matrix is a set of numbers that are aligned vertically. Each number is an entry, sometimes called an element, of the matrix. Matrices (plural) are enclosed in [ ] or ( ), and are usually named with capital letters. For example, three matrices named<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mtext/></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>C</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>are shown below.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 1 2 3 4 ],B=[ 1 2 7 0 −5 6 7 8 2 ],C=[ −1    0    3      3 2 1 ]

Describing Matrices

A matrix is often referred to by its size or dimensions:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>m</mi><mtext> </mtext><mo>×</mo><mtext> </mtext><mi>n</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>indicating<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>m</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>rows and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>n</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>columns. Matrix entries are defined first by row and then by column. For example, to locate the entry in matrix<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>identified as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> a ij ,we look for the entry in row<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>i</mi><mo>,</mo><mtext/></mrow></annotation-xml></semantics></math>column<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>j</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>In matrix<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext>,  </mtext></mrow></annotation-xml></semantics></math>shown below, the entry in row 2, column 3 is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> a 23 .

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ]

A square matrix is a matrix with dimensions<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>n</mi><mtext> </mtext><mo>×</mo><mtext> </mtext><mi>n</mi><mo>,</mo><mtext/></mrow></annotation-xml></semantics></math>meaning that it has the same number of rows as columns. The<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>3</mn><mo>×</mo><mn>3</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>matrix above is an example of a square matrix.

A row matrix is a matrix consisting of one row with dimensions<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>1</mn><mtext> </mtext><mo>×</mo><mtext> </mtext><mi>n</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> a 11 a 12 a 13 ]

A column matrix is a matrix consisting of one column with dimensions<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>m</mi><mtext> </mtext><mo>×</mo><mtext> </mtext><mn>1.</mn></mrow></annotation-xml></semantics></math>

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> a 11 a 21 a 31 ]

A matrix may be used to represent a system of equations. In these cases, the numbers represent the coefficients of the variables in the system. Matrices often make solving systems of equations easier because they are not encumbered with variables. We will investigate this idea further in the next section, but first we will look at basic matrix operations.

Matrices

A matrix is a rectangular array of numbers that is usually named by a capital letter:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>,</mo><mtext/></mrow></annotation-xml></semantics></math>and so on. Each entry in a matrix is referred to as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> a ij ,such that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>i</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>represents the row and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>j</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>represents the column. Matrices are often referred to by their dimensions:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>m</mi><mtext> </mtext><mo>×</mo><mtext> </mtext><mi>n</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>indicating<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>m</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>rows and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>n</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>columns.

Finding the Dimensions of the Given Matrix and Locating Entries

Given matrix<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mo>:</mo></mrow></annotation-xml></semantics></math>

  1. What are the dimensions of matrix<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mo>?</mo></mrow></annotation-xml></semantics></math>
  2. What are the entries at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> a 31  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> a 22 ?
    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 2 1 0 2 4 7 3 1 −2 ]
  1. The dimensions are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>3</mn><mtext> </mtext><mo>×</mo><mtext> </mtext><mn>3</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>because there are three rows and three columns.
  2. Entry<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> a 31  is the number at row 3, column 1, which is 3. The entry<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> a 22  is the number at row 2, column 2, which is 4. Remember, the row comes first, then the column.

Adding and Subtracting Matrices

We use matrices to list data or to represent systems. Because the entries are numbers, we can perform operations on matrices. We add or subtract matrices by adding or subtracting corresponding entries.

In order to do this, the entries must correspond. Therefore, addition and subtraction of matrices is only possible when the matrices have the same dimensions. We can add or subtract a<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>3</mn><mtext> </mtext><mo>×</mo><mtext> </mtext><mn>3</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>matrix and another<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>3</mn><mtext> </mtext><mo>×</mo><mtext> </mtext><mn>3</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>matrix, but we cannot add or subtract a<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mtext> </mtext><mo>×</mo><mtext> </mtext><mn>3</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>matrix and a<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>3</mn><mtext> </mtext><mo>×</mo><mtext> </mtext><mn>3</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>matrix because some entries in one matrix will not have a corresponding entry in the other matrix.

Adding and Subtracting Matrices

Given matrices<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>of like dimensions, addition and subtraction of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>will produce matrix<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>C</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>or

 
matrix<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>D</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>of the same dimension.

 

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>+</mo><mi>B</mi><mo>=</mo><mi>C</mi><mtext> such that </mtext><msub/></mrow></annotation-xml></semantics></math> a ij + b ij = c ij
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>−</mo><mi>B</mi><mo>=</mo><mi>D</mi><mtext> such that </mtext><msub/></mrow></annotation-xml></semantics></math> a ij − b ij = d ij

Matrix addition is commutative.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>+</mo><mi>B</mi><mo>=</mo><mi>B</mi><mo>+</mo><mi>A</mi></mrow></annotation-xml></semantics></math>

It is also associative.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> A+B )+C=A+( B+C )
Finding the Sum of Matrices

Find the sum of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>,</mo><mtext/></mrow></annotation-xml></semantics></math>given

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> a b c d ]   and  B=[ e f g h ]

Add corresponding entries.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>A</mi><mo>+</mo><mi>B</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> a b c d ]+[ e f g h ]          =[ a+e b+f c+g d+h ]
Adding Matrix A and Matrix B

Find the sum of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 4 1 3 2 ]  and  B=[ 5 9 0 7 ]

Add corresponding entries. Add the entry in row 1, column 1,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> a 11 ,of matrix<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>to the entry in row 1, column 1,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msub/></mrow></annotation-xml></semantics></math> b 11 ,of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>Continue the pattern until all entries have been added.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>A</mi><mo>+</mo><mi>B</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 4 1 3 2 ]+[ 5 9 0 7 ]          =[ 4+5 1+9 3+0 2+7 ]          =[ 9 10 3 9 ]
Finding the Difference of Two Matrices

Find the difference of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> −2 3 0 1 ]  and  B=[ 8 1 5 4 ]

We subtract the corresponding entries of each matrix.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>A</mi><mo>−</mo><mi>B</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> −2 3 0 1 ]−[ 8 1 5 4 ]          =[ −2−8 3−1 0−5 1−4 ]          =[ −10 2 −5 −3 ]
Finding the Sum and Difference of Two 3 x 3 Matrices

Given<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>:</mo></mrow></annotation-xml></semantics></math>

  1. Find the sum.
  2. Find the difference.
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 2 −10 −2 14 12 10 4 −2 2 ] and B=[ 6 10 −2 0 −12 −4 −5 2 −2 ]
  1. Add the corresponding entries.
    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>A</mi><mo>+</mo><mi>B</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 2   −10   −2 14   12   10 4   −2   2 ]+[ 6   10   −2 0   −12   −4 −5   2   −2 ]                =[ 2+6   −10+10   −2−2 14+0  12−12   10−4 4−5   −2+2   2−2 ]                =[ 8   0   −4 14   0   6 −1   0   0 ]
  2. Subtract the corresponding entries.
    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>A</mi><mo>−</mo><mi>B</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 2 −10 −2 14 12 10 4 −2 2 ]−[ 6 10 −2 0 −12 −4 −5 2 −2 ]                =[ 2−6   −10−10   −2+2 14−0   12+12   10+44+5   −2−2   2+2 ]                =[ −4   −20   0 14   24   14 9   −4   4 ]

Add matrix<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and matrix<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 2 6 1 0 1 −3 ]  and  B=[ 3 −2 1 5 −4 3 ]
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>+</mo><mi>B</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 2 1 1     6 ​​​     0    −3 ]+[    3  1 −4   −2      5       3 ]=[ 2  + 3 1   +   1 1+(−4)        6+(−2) 0  +  5 −3   +   3 ]=[  5   2 −3        4 5 0]

Finding Scalar Multiples of a Matrix

Besides adding and subtracting whole matrices, there are many situations in which we need to multiply a matrix by a constant called a scalar. Recall that a scalar is a real number quantity that has magnitude, but not direction. For example, time, temperature, and distance are scalar quantities. The process of scalar multiplication involves multiplying each entry in a matrix by a scalar. A scalar multiple is any entry of a matrix that results from scalar multiplication.

Consider a real-world scenario in which a university needs to add to its inventory of computers, computer tables, and chairs in two of the campus labs due to increased enrollment. They estimate that 15% more equipment is needed in both labs. The school’s current inventory is displayed in [link].

  Lab A Lab B
Computers 15 27
Computer Tables 16 34
Chairs 16 34

Converting the data to a matrix, we have

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>C</mi></msub></mrow></annotation-xml></semantics></math> 2013 =[ 15 16 16         27 34 34 ]

To calculate how much computer equipment will be needed, we multiply all entries in matrix<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>C</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>by 0.15.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo stretchy="false">(</mo><mn>0.15</mn><mo stretchy="false">)</mo><msub/></mrow></annotation-xml></semantics></math> C 2013 =[ (0.15)15 (0.15)16 (0.15)16          (0.15)27 (0.15)34 (0.15)34 ]=[ 2.25 2.4 2.4       4.05 5.1 5.1 ]

We must round up to the next integer, so the amount of new equipment needed is

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 3 3 3       5 6 6 ]

Adding the two matrices as shown below, we see the new inventory amounts.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 15 16 16         27 34 34 ]+[ 3 3 3       5 6 6 ]=[ 18 19 19       32 40 40 ]

This means

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msub><mi>C</mi></msub></mrow></annotation-xml></semantics></math> 2014 =[ 18 19 19       32 40 40 ]

Thus, Lab A will have 18 computers, 19 computer tables, and 19 chairs; Lab B will have 32 computers, 40 computer tables, and 40 chairs.

Scalar Multiplication

Scalar multiplication involves finding the product of a constant by each entry in the matrix. Given

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> a 11 a 12 a 21 a 22 ]

the scalar multiple<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>c</mi><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>c</mi><mi>A</mi><mo>=</mo><mi>c</mi><mrow><mo>[</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> a 11 a 12 a 21 a 22 ]     =[ c a 11 c a 12 c a 21 c a 22 ]

Scalar multiplication is distributive. For the matrices<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>C</mi><mtext> </mtext></mrow></annotation-xml></semantics></math> with scalars<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>a</mi><mtext>  </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>b</mi><mo>,</mo></mrow></annotation-xml></semantics></math>

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable columnalign="left"><mtr><mtd><mrow/></mtd></mtr><mtr><mtd><mtable><mtr><mtd><mrow><mi>a</mi><mo stretchy="false">(</mo><mi>A</mi><mo>+</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mi>A</mi><mo>+</mo><mi>a</mi><mi>B</mi></mrow></mtd></mtr></mtable></mtd></mtr></mtable></annotation-xml></semantics></math> (a+b)A=aA+bA
Multiplying the Matrix by a Scalar

Multiply matrix<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>by the scalar 3.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 8 1 5 4 ]

Multiply each entry in<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>by the scalar 3.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mn>3</mn><mi>A</mi><mo>=</mo><mn>3</mn><mrow><mo>[</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 8   1 5   4 ]         = [ 3⋅8   3⋅1 3⋅5   3⋅4 ]         = [ 24 3 15 12 ]

Given matrix<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>,</mo><mtext/></mrow></annotation-xml></semantics></math>find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>−2</mn><mi>B</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>where

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>B</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 4 1 3 2 ]

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>−2</mn><mi>B</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> −8 −2 −6 −4 ]

Finding the Sum of Scalar Multiples

Find the sum<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>3</mn><mi>A</mi><mo>+</mo><mn>2</mn><mi>B</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 1 −2 0 0 −1 2 4 3 −6 ] and B=[ −1 2 1 0 −3 2 0 1 −4 ]

First, find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>3</mn><mi>A</mi><mo>,</mo><mtext/></mrow></annotation-xml></semantics></math>then<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>B</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

 

 

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mn>3</mn><mi>A</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 3⋅1   3(−2)   3⋅0 3⋅0   3(−1)   3⋅2 3⋅4   3⋅3   3(−6) ]        =[ 3   −6   0 0   −3   6 12   9   −18 ]
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mn>2</mn><mi>B</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></mtd></mtr></mtable></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 2(−1)   2⋅2   2⋅1 2⋅0   2(−3)   2⋅2 2⋅0   2⋅1   2(−4) ]        =[ −2 4 2 0 −6 4 0 2 −8 ]

Now, add<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>3</mn><mi>A</mi><mo>+</mo><mn>2</mn><mi>B</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mn>3</mn><mi>A</mi><mo>+</mo><mn>2</mn><mi>B</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 3 −6 0 0 −3 6 12 9 −18 ]+[ −2 4 2 0 −6 4 0 2 −8 ]              =[ 3−2   −6+4 0+2 0+0   −3−6 6+4 12+0   9+2 −18−8 ]             =[ 1   −2 2 0   −9 10 12   11 −26 ]

Finding the Product of Two Matrices

In addition to multiplying a matrix by a scalar, we can multiply two matrices. Finding the product of two matrices is only possible when the inner dimensions are the same, meaning that the number of columns of the first matrix is equal to the number of rows of the second matrix. If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is an<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>m</mi><mtext> </mtext><mo>×</mo><mtext> </mtext><mi>r</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>matrix and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is an<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>r</mi><mtext> </mtext><mo>×</mo><mtext> </mtext><mi>n</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>matrix, then the product matrix<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mi>B</mi><mtext>  </mtext></mrow></annotation-xml></semantics></math>is an<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>m</mi><mtext> </mtext><mo>×</mo><mtext> </mtext><mi>n</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>matrix. For example, the product<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mi>B</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is possible because the number of columns in<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is the same as the number of rows in<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>If the inner dimensions do not match, the product is not defined.

We multiply entries of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>with entries of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>according to a specific pattern as outlined below. The process of matrix multiplication becomes clearer when working a problem with real numbers.

To obtain the entries in row<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>i</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mi>B</mi><mo>,</mo><mtext/></mrow></annotation-xml></semantics></math>we multiply the entries in row<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>i</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math> by column<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>j</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>in<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and add. For example, given matrices<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math> and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>,</mo><mtext/></mrow></annotation-xml></semantics></math>where the dimensions of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mtext> </mtext><mo>×</mo><mtext> </mtext><mn>3</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>and the dimensions of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>3</mn><mtext> </mtext><mo>×</mo><mtext> </mtext><mn>3</mn><mo>,</mo><mtext/></mrow></annotation-xml></semantics></math>the product of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mi>B</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>will be a<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mtext> </mtext><mo>×</mo><mtext> </mtext><mn>3</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>matrix.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> a 11 a 12 a 13 a 21 a 22 a 23 ] and B=[ b 11 b 12 b 13 b 21 b 22 b 23 b 31 b 32 b 33 ]

Multiply and add as follows to obtain the first entry of the product matrix<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mi>B</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

  1. To obtain the entry in row 1, column 1 of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mi>B</mi><mo>,</mo><mtext/></mrow></annotation-xml></semantics></math>multiply the first row in<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>by the first column in<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>,</mo></mrow></annotation-xml></semantics></math>and add.
    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> a 11 a 12 a 13 ]⋅[ b 11 b 21 b 31 ]= a 11 ⋅ b 11 + a 12 ⋅ b 21 + a 13 ⋅ b 31
  2. To obtain the entry in row 1, column 2 of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mi>B</mi><mo>,</mo><mtext/></mrow></annotation-xml></semantics></math>multiply the first row of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math> by the second column in<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>,</mo></mrow></annotation-xml></semantics></math>and add.
    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> a 11 a 12 a 13 ]⋅[ b 12 b 22 b 32 ]= a 11 ⋅ b 12 + a 12 ⋅ b 22 + a 13 ⋅ b 32
  3. To obtain the entry in row 1, column 3 of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mi>B</mi><mo>,</mo><mtext/></mrow></annotation-xml></semantics></math>multiply the first row of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>by the third column in<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>,</mo></mrow></annotation-xml></semantics></math>and add.
    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> a 11 a 12 a 13 ]⋅[ b 13 b 23 b 33 ]= a 11 ⋅ b 13 + a 12 ⋅ b 23 + a 13 ⋅ b 33

We proceed the same way to obtain the second row of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mi>B</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>In other words, row 2 of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>times column 1 of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>row 2 of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>times column 2 of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>row 2 of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>times column 3 of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>When complete, the product matrix will be

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mi>B</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> a 11 ⋅ b 11 + a 12 ⋅ b 21 + a 13 ⋅ b 31 a 21 ⋅ b 11 + a 22 ⋅ b 21 + a 23 ⋅ b 31           a 11 ⋅ b 12 + a 12 ⋅ b 22 + a 13 ⋅ b32 a 21 ⋅ b 12 + a 22 ⋅ b 22 + a 23 ⋅ b 32          a 11 ⋅ b 13 + a 12 ⋅ b 23 + a 13 ⋅ b 33 a 21 ⋅ b 13 + a 22 ⋅ b 23 + a 23 ⋅ b 33 ]
Properties of Matrix Multiplication

For the matrices<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mtext/></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>C</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>the following properties hold.

  • Matrix multiplication is associative:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> AB )C=A( BC ).
  • Matrix multiplication is distributive:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mtable columnalign="left"><mtr><mtd><mrow/></mtd></mtr><mtr><mtd><mtext> </mtext><mi>C</mi><mo stretchy="false">(</mo><mi>A</mi><mo>+</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mi>C</mi><mi>A</mi><mo>+</mo><mi>C</mi><mi>B</mi><mo>,</mo></mtd></mtr></mtable></mtd></mtr></mtable></mrow></annotation-xml></semantics></math>  (A+B)C=AC+BC.

Note that matrix multiplication is not commutative.

Multiplying Two Matrices

Multiply matrix<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and matrix<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 1 2 3 4 ]  and  B=[ 5 6 7 8 ]

First, we check the dimensions of the matrices. Matrix<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>has dimensions<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mtext> </mtext><mo>×</mo><mtext> </mtext><mn>2</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>and matrix<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>has dimensions<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mtext> </mtext><mo>×</mo><mtext> </mtext><mn>2.</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>The inner dimensions are the same so we can perform the multiplication. The product will have the dimensions<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mtext> </mtext><mo>×</mo><mtext> </mtext><mn>2.</mn></mrow></annotation-xml></semantics></math>

We perform the operations outlined previously.

Multiplying Two Matrices

Given<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>:</mo></mrow></annotation-xml></semantics></math>

  1. Find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mi>B</mi><mo>.</mo></mrow></annotation-xml></semantics></math>
  2. Find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mi>A</mi><mo>.</mo></mrow></annotation-xml></semantics></math>
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> −1 2 3    4 0 5 ] and  B=[   5 −4   2     −1   0   3 ]
  1. As the dimensions of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mtext> </mtext><mo>×</mo><mtext> </mtext><mn>3</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>and the dimensions of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>3</mn><mtext> </mtext><mo>×</mo><mtext> </mtext><mn>2</mn><mo>,</mo><mtext/></mrow></annotation-xml></semantics></math>these matrices can be multiplied together because the number of columns in<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>matches the number of rows in<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>The resulting product will be a<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mtext> </mtext><mo>×</mo><mtext> </mtext><mn>2</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>matrix, the number of rows in<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>by the number of columns in<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>.</mo></mrow></annotation-xml></semantics></math>
    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>A</mi><mi>B</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> −1 2 3 4 0 5 ]   [ 5 −1 −4 0 2 3 ]      =[ −1(5)+2(−4)+3(2)   −1(−1)+2(0)+3(3) 4(5)+0(−4)+5(2)   4(−1)+0(0)+5(3)]      =[ −7 10 30 11 ]
  2. The dimensions of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>3</mn><mo> × </mo><mn>2</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>and the dimensions of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mo> × </mo><mn>3.</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>The inner dimensions match so the product is defined and will be a<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>3</mn><mo> × </mo><mn>3</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>matrix.
    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>B</mi><mi>A</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 5 −1 −4 0 2 3 ]   [ −1 2 3 4 0 5 ]      =[ 5(−1)+−1(4)   5(2)+−1(0)   5(3)+−1(5) −4(−1)+0(4)   −4(2)+0(0)   −4(3)+0(5) 2(−1)+3(4)   2(2)+3(0)   2(3)+3(5) ]      =[ −9 10 10 4 −8 −12 10 4 21 ]
Analysis

Notice that the products<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mi>B</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>are not equal.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mi>B</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> −7 10 30 11 ]≠[ −9 10 10 4 −8 −12 10 4 21 ]=BA

This illustrates the fact that matrix multiplication is not commutative.

Is it possible for AB to be defined but not BA?

Yes, consider a matrix A with dimension<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>3</mn><mtext> </mtext><mo>×</mo><mtext> </mtext><mn>4</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>and matrix B with dimension<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>4</mn><mtext> </mtext><mo>×</mo><mtext> </mtext><mn>2.</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>For the product AB the inner dimensions are 4 and the product is defined, but for the product BA the inner dimensions are 2 and 3 so the product is undefined.

Using Matrices in Real-World Problems

Let’s return to the problem presented at the opening of this section. We have [link], representing the equipment needs of two soccer teams.

  Wildcats Mud Cats
Goals 6 10
Balls 30 24
Jerseys 14 20

We are also given the prices of the equipment, as shown in [link].

Goal $300
Ball $10
Jersey $30

We will convert the data to matrices. Thus, the equipment need matrix is written as

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>E</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 6 30 14       10 24 20 ]

The cost matrix is written as

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>C</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 300 10 30 ]

We perform matrix multiplication to obtain costs for the equipment.


 

 

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow/></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>C</mi><mi>E</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 300 10 30 ]⋅[ 6 10 30 24 14 20 ]      =[ 300(6)+10(30)+30(14) 300(10)+10(24)+30(20) ]      =[ 2,520 3,840 ]

The total cost for equipment for the Wildcats is $2,520, and the total cost for equipment for the Mud Cats is $3,840.

Given a matrix operation, evaluate using a calculator.

  1. Save each matrix as a matrix variable<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> A ],[ B ],[ C ],...
  2. Enter the operation into the calculator, calling up each matrix variable as needed.
  3. If the operation is defined, the calculator will present the solution matrix; if the operation is undefined, it will display an error message.
Using a Calculator to Perform Matrix Operations

Find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mi>B</mi><mo>−</mo><mi>C</mi><mtext> </mtext></mrow></annotation-xml></semantics></math> given

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> −15 25 32 41 −7 −28 10 34 −2 ],B=[ 45 21 −37 −24 52 19 6 −48 −31 ],and C=[ −100 −89 −98 25 −56 74 −67 42 −75 ].

On the matrix page of the calculator, we enter matrix<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>above as the matrix variable<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> A ],matrix<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>above as the matrix variable<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> B ],and matrix<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>C</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>above as the matrix variable<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> C ].

On the home screen of the calculator, we type in the problem and call up each matrix variable as needed.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> A ]×[ B ]−[ C ]

The calculator gives us the following matrix.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> −983   −462   136 1,820   1,897   −856 −311   2,032   413 ]

Access these online resources for additional instruction and practice with matrices and matrix operations.

Key Concepts

  • A matrix is a rectangular array of numbers. Entries are arranged in rows and columns.
  • The dimensions of a matrix refer to the number of rows and the number of columns. A<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>3</mn><mo>×</mo><mn>2</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>matrix has three rows and two columns. See [link].
  • We add and subtract matrices of equal dimensions by adding and subtracting corresponding entries of each matrix. See[link], [link], [link], and [link].
  • Scalar multiplication involves multiplying each entry in a matrix by a constant. See [link].
  • Scalar multiplication is often required before addition or subtraction can occur. See [link].
  • Multiplying matrices is possible when inner dimensions are the same—the number of columns in the first matrix must match the number of rows in the second.
  • The product of two matrices,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>,</mo></mrow></annotation-xml></semantics></math>is obtained by multiplying each entry in row 1 of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>by each entry in column 1 of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>then multiply each entry of row 1 of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>by each entry in columns 2 of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>,</mo><mtext/></mrow></annotation-xml></semantics></math>and so on. See [link] and [link].
  • Many real-world problems can often be solved using matrices. See [link].
  • We can use a calculator to perform matrix operations after saving each matrix as a matrix variable. See [link].

Section Exercises

Verbal

Can we add any two matrices together? If so, explain why; if not, explain why not and give an example of two matrices that cannot be added together.

No, they must have the same dimensions. An example would include two matrices of different dimensions. One cannot add the following two matrices because the first is a<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mo>×</mo><mn>2</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>matrix and the second is a<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mo>×</mo><mn>3</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>matrix.<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 1 2 3 4 ]+[ 6 5 4 3 2 1 ] has no sum.

Can we multiply any column matrix by any row matrix? Explain why or why not.

Can both the products<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mi>B</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>be defined? If so, explain how; if not, explain why.

Yes, if the dimensions of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>m</mi><mo>×</mo><mi>n</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and the dimensions of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>are<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>n</mi><mo>×</mo><mi>m</mi><mo>,</mo><mtext/></mrow></annotation-xml></semantics></math>both products will be defined.

Can any two matrices of the same size be multiplied? If so, explain why, and if not, explain why not and give an example of two matrices of the same size that cannot be multiplied together.

Does matrix multiplication commute? That is, does<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mi>B</mi><mo>=</mo><mi>B</mi><mi>A</mi><mo>?</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>If so, prove why it does. If not, explain why it does not.

Not necessarily. To find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mi>B</mi><mo>,</mo><mtext/></mrow></annotation-xml></semantics></math>we multiply the first row of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>by the first column of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>to get the first entry of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mi>B</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>To find<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mi>A</mi><mo>,</mo><mtext/></mrow></annotation-xml></semantics></math>we multiply the first row of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>by the first column of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>to get the first entry of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mi>A</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>Thus, if those are unequal, then the matrix multiplication does not commute.

Algebraic

For the following exercises, use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 1 3 0 7 ],B=[ 2 14 22 6 ],C=[ 1 5 8 92 12 6 ],D=[ 10 14 7 2 5 61 ],E=[ 6 12 14 5 ],F=[ 0 9 78 17 15 4 ]

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>+</mo><mi>B</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>C</mi><mo>+</mo><mi>D</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 11 19 15 94 17 67 ]

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>+</mo><mi>C</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>B</mi><mo>−</mo><mi>E</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> −4 2 8 1 ]

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>C</mi><mo>+</mo><mi>F</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>D</mi><mo>−</mo><mi>B</mi></mrow></annotation-xml></semantics></math>

Undidentified; dimensions do not match

For the following exercises, use the matrices below to perform scalar multiplication.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 4 6 13 12 ],B=[ 3 9 21 12 0 64 ],C=[ 16 3 7 18 90 5 3 29 ],D=[ 18 12 13 8 14 6 7 4 21 ]

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>5</mn><mi>A</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>3</mn><mi>B</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 9 27 63 36 0 192 ]

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>−2</mn><mi>B</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>−4</mn><mi>C</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> −64 −12 −28 −72 −360 −20 −12 −116 ]

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>1</mn></mfrac></mrow></annotation-xml></semantics></math> 2 C

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>100</mn><mi>D</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 1,800 1,200 1,300 800 1,400 600 700 400 2,100 ]

For the following exercises, use the matrices below to perform matrix multiplication.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> −1 5 3 2 ],B=[ 3 6 4 −8 0 12 ],C=[ 4 10 −2 6 5 9 ],D=[ 2 −3 12 9 3 1 0 8 −10 ]

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mi>B</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>B</mi><mi>C</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 20 102 28 28 ]

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>C</mi><mi>A</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>B</mi><mi>D</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 60 41 2 −16 120 −216 ]

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>D</mi><mi>C</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>C</mi><mi>B</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> −68 24 136 −54 −12 64 −57 30 128 ]

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 2 −5 6 7 ],B=[ −9 6 −4 2 ],C=[ 0 9 7 1 ],D=[ −8 7 −5 4 3 2 0 9 2 ],E=[ 4 5 3 7 −6 −5 1 0 9 ]

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>+</mo><mi>B</mi><mo>−</mo><mi>C</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>4</mn><mi>A</mi><mo>+</mo><mn>5</mn><mi>D</mi></mrow></annotation-xml></semantics></math>

Undefined; dimensions do not match.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>2</mn><mi>C</mi><mo>+</mo><mi>B</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>3</mn><mi>D</mi><mo>+</mo><mn>4</mn><mi>E</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> −8 41 −3 40 −15 −14 4 27 42 ]

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>C</mi><mn>−0.5</mn><mi>D</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mn>100</mn><mi>D</mi><mn>−10</mn><mi>E</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> −840 650 −530 330 360 250 −10 900 110 ]

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> A 2 =A⋅A )

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> −10 20 5 25 ],B=[ 40 10 −20 30 ],C=[ −1 0 0 −1 1 0 ]

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mi>B</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>B</mi><mi>A</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> −350 1,050 350 350 ]

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>C</mi><mi>A</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>B</mi><mi>C</mi></mrow></annotation-xml></semantics></math>

Undefined; inner dimensions do not match.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>A</mi></msup></mrow></annotation-xml></semantics></math> 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>B</mi></msup></mrow></annotation-xml></semantics></math> 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 1,400 700 −1,400 700 ]

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>C</mi></msup></mrow></annotation-xml></semantics></math> 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>B</mi></msup></mrow></annotation-xml></semantics></math> 2 A 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 332,500 927,500 −227,500 87,500 ]

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>A</mi></msup></mrow></annotation-xml></semantics></math> 2 B 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mo stretchy="false">(</mo><mi>A</mi><mi>B</mi><mo stretchy="false">)</mo></mrow></msup></mrow></annotation-xml></semantics></math> 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 490,000 0 0 490,000 ]

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mrow><mo stretchy="false">(</mo><mi>B</mi><mi>A</mi><mo stretchy="false">)</mo></mrow></msup></mrow></annotation-xml></semantics></math> 2

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> A 2 =A⋅A )

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 1 0 2 3 ],B=[ −2 3 4 −1 1 −5 ],C=[ 0.5 0.1 1 0.2 −0.5 0.3 ],D=[ 1 0 −1 −6 7 5 4 2 1 ]

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mi>B</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> −2 3 4 −7 9 −7 ]

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>B</mi><mi>A</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>B</mi><mi>D</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> −4 29 21 −27 −3 1 ]

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>D</mi><mi>C</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>D</mi></msup></mrow></annotation-xml></semantics></math> 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> −3 −2 −2 −28 59 46 −4 16 7 ]

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>A</mi></msup></mrow></annotation-xml></semantics></math> 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>D</mi></msup></mrow></annotation-xml></semantics></math> 3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 1 −18 −9 −198 505 369 −72 126 91 ]

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo stretchy="false">(</mo><mi>A</mi><mi>B</mi><mo stretchy="false">)</mo><mi>C</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo stretchy="false">(</mo><mi>B</mi><mi>C</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 0 1.6 9 −1 ]

Technology

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. Use a calculator to verify your solution.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> −2 0 9 1 8 −3 0.5 4 5 ],B=[ 0.5 3 0 −4 1 6 8 7 2 ],C=[ 1 0 1 0 1 0 1 0 1 ]

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mi>B</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>B</mi><mi>A</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 2 24 −4.5 12 32 −9 −8 64 61 ]

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>C</mi><mi>A</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>B</mi><mi>C</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 0.5 3 0.5 2 1 2 10 7 10 ]

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mi>B</mi><mi>C</mi></mrow></annotation-xml></semantics></math>

Extensions

For the following exercises, use the matrix below to perform the indicated operation on the given matrix.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>B</mi><mo>=</mo><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 1 0 0 0 0 1 0 1 0 ]

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>B</mi></msup></mrow></annotation-xml></semantics></math> 2

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 1 0 0 0 1 0 0 0 1 ]

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>B</mi></msup></mrow></annotation-xml></semantics></math> 3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>B</mi></msup></mrow></annotation-xml></semantics></math> 4

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 1 0 0 0 1 0 0 0 1 ]

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>B</mi></msup></mrow></annotation-xml></semantics></math> 5

Using the above questions, find a formula for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> B n . Test the formula for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> B 201  and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><msup/></mrow></annotation-xml></semantics></math> B 202 ,using a calculator.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><msup><mi>B</mi></msup></mrow></annotation-xml></semantics></math> n ={ [ 1 0 0 0 1 0 0 0 1 ], n even, [ 1 0 0 0 0 1 0 1 0 ], n odd.

Glossary

column
a set of numbers aligned vertically in a matrix
entry
an element, coefficient, or constant in a matrix
matrix
a rectangular array of numbers
row
a set of numbers aligned horizontally in a matrix
scalar multiple
an entry of a matrix that has been multiplied by a scalar