4.3: Matrix Multiplication
( \newcommand{\kernel}{\mathrm{null}\,}\)
Compatible Matrices
We are going to multiply together two matrices, one of size
Notice the number of columns of the leftmost matrix is equal to the number of rows of the rightmost matrix.
For the product,
Matrices as Collections of Row and Column Matrices
It is productive to think of a matrix as a collection of individual row matrices and column matrices.For example, we can think of the matrix
- the three row matrices,
and and - the two column matrices
and .
(If you need a review of row and column matrices, see Section 4.2)
Multiplication of Two Matrices
To multiply two compatible matrices
Suppose the size of matrix
Some of the entries of the product
1st row of matrix
1st row of matrix
2nd row of matrix
3rd row of matrix
3rd row of matrix
Do you see the general rule for producing any particular entry?
To get the entry in row
the
Compute the product of the matrices
First note that the two matrices are compatible
Solution
The product is the
Since we are multiplying 3 rows through 2 columns, there will be 6 entries. The six entries of
So,
Your Turn: Show that the product of the matrices
Using Technology
You can see that multiplying matrices together involves a lot of arithmetic and can be cumbersome. We can use technology to help us through the process.
Go to www.wolframalpha.com.
To find the product of the two matrices of above Your Turn Example, enter [[2,3], [4,1]] * [[2,3,0], [1,2,4]] in the entry field. WolframAlpha sees a matrix as a collection of row matrices.
Both entries and rows are separated by commas and WA does not see spaces.
Wolframalpha tells you what it thinks you entered, then tells you its answer
Try these
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Compare your answers to question 1 and 2. If you got them right, would you say that matrix multiplication is or is not commutative?
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Is not commutative
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Not defined