Chapter 8, Problem 37RE

a.

To determine

Sum of matrix A+B

Answer to Problem 37RE

Sum of matrix is;

  $\left[\begin{array}{l}-1\text{ 8}\\ \text{15 13}\end{array}\right]$

Explanation of Solution

Given:

Matrices are given as;

  $A=\left[\begin{array}{l}2\text{ -2}\\ \text{3 5}\end{array}\right],B=\left[\begin{array}{l}-3\text{ 10}\\ \text{12 8}\end{array}\right]$

Dimension of matrix A is $2\times 2$ and size of matrix B is $2\times 2$ both are square matrix therefore sum of the matrix is possible.

And size of matrix A+B will be $2\times 2$

Therefore. sum of matrix,

  $\begin{array}{l}A+B=\left[\begin{array}{l}2\text{ -2}\\ \text{3 5}\end{array}\right]+\left[\begin{array}{l}-3\text{ 10}\\ \text{12 8}\end{array}\right]\\ =\left[\begin{array}{l}2+\left(-3\right)\text{ -2+10}\\ \text{3+12 5+8 }\end{array}\right]\\ =\left[\begin{array}{l}\text{5 -12}\\ \text{15 13}\end{array}\right]\end{array}$

b.

To determine

Subtraction of matrix $A-B$

Answer to Problem 37RE

Subtraction of matrix is;

  $\left[\begin{array}{l}\text{5 -12}\\ \text{-9 -3}\end{array}\right]$

Explanation of Solution

Given:

Matrices are given as;

  $A=\left[\begin{array}{l}2\text{ -2}\\ \text{3 5}\end{array}\right],B=\left[\begin{array}{l}-3\text{ 10}\\ \text{12 8}\end{array}\right]$

Dimension of matrix A is $2\times 2$ and size of matrix B is $2\times 2$ both are square matrix therefore subtraction of the matrix is possible.

And size of matrix A-B will be $2\times 2$

Therefore. subtraction of matrix,

  $\begin{array}{l}A-B=\left[\begin{array}{l}2\text{ -2}\\ \text{3 5}\end{array}\right]-\left[\begin{array}{l}-3\text{ 10}\\ \text{12 8}\end{array}\right]\\ =\left[\begin{array}{l}2-\left(-3\right)\text{ -2-10}\\ \text{3-12 5-8 }\end{array}\right]\\ =\left[\begin{array}{l}\text{5 -12}\\ \text{-9 -3}\end{array}\right]\end{array}$

c.

To determine

Expression of matrix 4A

Answer to Problem 37RE

Matrix 4A is;

  $\left[\begin{array}{l}8\text{ -8}\\ \text{12 20}\end{array}\right]$

Explanation of Solution

Given:

Matrices are given as;

  $A=\left[\begin{array}{l}2\text{ -2}\\ \text{3 5}\end{array}\right],B=\left[\begin{array}{l}-3\text{ 10}\\ \text{12 8}\end{array}\right]$

  $\begin{array}{l}4A=4\left[\begin{array}{l}2\text{ -2}\\ \text{3 5}\end{array}\right]\\ =\left[\begin{array}{l}4\times 2\text{ 4}\times \text{-2}\\ 4\times 3\text{ 4}\times \text{5}\end{array}\right]\\ =\left[\begin{array}{l}8\text{ -8}\\ \text{12 20}\end{array}\right]\end{array}$

d.

To determine

Expression of matrix A+3B

Answer to Problem 37RE

Matrix A+3B is;

  $\left[\begin{array}{l}-7\text{ 28}\\ \text{39 29}\end{array}\right]$

Explanation of Solution

Given:

Matrices are given as;

  $A=\left[\begin{array}{l}2\text{ -2}\\ \text{3 5}\end{array}\right],B=\left[\begin{array}{l}-3\text{ 10}\\ \text{12 8}\end{array}\right]$

  $\begin{array}{l}A+3B=\left[\begin{array}{l}2\text{ -2}\\ \text{3 5}\end{array}\right]+3\left[\begin{array}{l}-3\text{ 10}\\ \text{12 8}\end{array}\right]\\ =\left[\begin{array}{l}2\text{ -2}\\ \text{3 5}\end{array}\right]+\left[\begin{array}{l}-9\text{ 30}\\ \text{36 24}\end{array}\right]\\ =\left[\begin{array}{l}-7\text{ 28}\\ \text{39 29}\end{array}\right]\end{array}$