3.2E: Exercises - Matrices and Matrix Operations

Last updated
Save as PDF

Page ID: 49624

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

Matrices and Matrix Operations

For the exercises 1-6, use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined.

\[A=\begin{bmatrix} 1 & 3\\ 0 & 7 \end{bmatrix}, B=\begin{bmatrix} 2 & 14\\ 22 & 6 \end{bmatrix}, C=\begin{bmatrix} 1 & 5\\ 8 & 92\\ 12 & 6 \end{bmatrix}, D=\begin{bmatrix} 10 & 14\\ 7 & 2\\ 5 & 61 \end{bmatrix}, E=\begin{bmatrix} 6 & 12\\ 14 & 5 \end{bmatrix}, F=\begin{bmatrix} 0 & 9\\ 78 & 17\\ 15 & 4 \end{bmatrix} \nonumber\]

1) \(A+B\)

2) \(C+D\)

Answer: \(\begin{bmatrix} 11 & 19\\ 15 & 94\\ 17 & 67 \end{bmatrix}\)

3) \(A+C\)

4) \(B-E\)

Answer: \(\begin{bmatrix} -4 & 2\\ 8 & 1 \end{bmatrix}\)

5) \(C+F\)

6) \(D-B\)

Answer: Undefined; dimensions do not match

For the exercises 7-12, use the matrices below to perform scalar multiplication.

\[A=\begin{bmatrix} 4 & 6\\ 13 & 12 \end{bmatrix}, B=\begin{bmatrix} 3 & 9\\ 21 & 12\\ 0 & 64 \end{bmatrix}, C=\begin{bmatrix} 16 & 3 & 7 & 18\\ 90 & 5 & 3 & 29 \end{bmatrix}, D=\begin{bmatrix} 18 & 12 & 13\\ 8 & 14 & 6\\ 7 & 4 & 21 \end{bmatrix} \nonumber\]

7) \(5A\)

8) \(3B\)

Answer: \(\begin{bmatrix} 9 & 27\\ 63 & 36\\ 0 & 192 \end{bmatrix}\)

9) \(-2B\)

10) \(-4C\)

Answer: \(\begin{bmatrix} -64 & -12 & -28 & -72\\ -360 & -20 & -12 & -116 \end{bmatrix}\)

11) \(\dfrac{1}{2}C\)

12) \(100D\)

Answer: \(\begin{bmatrix} 1,800 & 1,200 & 1,300\\ 800 & 1,400 & 600\\ 700 & 400 & 2,100 \end{bmatrix}\)

For the exercises 13-18, use the matrices below to perform matrix multiplication.

\[A=\begin{bmatrix} -1 & 5\\ 3 & 2 \end{bmatrix}, B=\begin{bmatrix} 3 & 6 & 4\\ -8 & 0 & 12 \end{bmatrix}, C=\begin{bmatrix} 4 & 10\\ -2 & 6\\ 5 & 9 \end{bmatrix}, D=\begin{bmatrix} 2 & -3 & 12\\ 9 & 3 & 1\\ 0 & 8 & -10 \end{bmatrix} \nonumber\]

13) \(AB\)

14) \(BC\)

Answer: \(\begin{bmatrix} 20 & 102\\ 28 & 28 \end{bmatrix}\)

15) \(CA\)

16) \(BD\)

Answer: \(\begin{bmatrix} 60 & 41 & 2\\ -16 & 120 & -216 \end{bmatrix}\)

17) \(DC\)

18) \(CB\)

Answer: \(\begin{bmatrix} -68 & 24 & 136\\ -54 & -12 & 64\\ -57 & 30 & 128 \end{bmatrix}\)

For the exercises 19-24, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed.

\[A=\begin{bmatrix} 2 & -5\\ 6 & 7 \end{bmatrix}, B=\begin{bmatrix} -9 & 6\\ -4 & 2 \end{bmatrix}, C=\begin{bmatrix} 0 & 9\\ 7 & 1 \end{bmatrix}, D=\begin{bmatrix} -8 & 7 & -5\\ 4 & 3 & 2\\ 0 & 9 & 2 \end{bmatrix}, E=\begin{bmatrix} 4 & 5 & 3\\ 7 & -6 & -5\\ 1 & 0 & 9 \end{bmatrix} \nonumber\]

19) \(A+B-C\)

20) \(4A+5D\)

Answer: Undefined; dimensions do not match.

21) \(2C+B\)

22) \(3D+4E\)

Answer: \(\begin{bmatrix} -8 & 41 & -3\\ 40 & -15 & -14\\ 4 & 27 & 42 \end{bmatrix}\)

23) \(C-0.5D\)

24) \(100D-10E\)

Answer: \(\begin{bmatrix} -840 & 650 & -530\\ 330 & 360 & 250\\ -10 & 900 & 110 \end{bmatrix}\)

Contributors and Attributions

Jay Abramson (Arizona State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at https://openstax.org/details/books/precalculus.