Chapter 8, Problem 57RE

a.

To determine

Product of matrix AB

Answer to Problem 57RE

Product of matrix is ;

  $\left[\begin{array}{l}-1\text{ -1}\\ \text{18 -4}\end{array}\right]$

Explanation of Solution

Given:

Matrices are given as;

  $A=\left[\begin{array}{l}1\text{ 3}\\ \text{4 1}\end{array}\right],B=\left[\begin{array}{l}5\text{ -1}\\ \text{-2 0}\end{array}\right]$

Dimension of matrix A is $2\times 2$ and size of matrix B is $2\times 2$ as the number of column of matrix A is equal to number of row of matrix B therefore product AB is possible.

And size of matrix AB will be $2\times 2$

Therefore. product of matrix

  $\begin{array}{l}AB=\left[\begin{array}{l}1\text{ 3}\\ \text{4 1}\end{array}\right]\left[\begin{array}{l}5\text{ -1}\\ \text{-2 0}\end{array}\right]\\ =\left[\begin{array}{l}1\times 5+3\times \left(-2\right)\text{ 1}\times \left(-1\right)+3\times 0\\ 4\times 5+1\times \left(-2\right)\text{ 1}\times \left(-1\right)+3\times 0\end{array}\right]\\ =\left[\begin{array}{l}-1\text{ -1}\\ \text{18 -4}\end{array}\right]\end{array}$

b.

To determine

Product of matrix BA

Answer to Problem 57RE

Product of matrix BA is;

  $\left[\begin{array}{l}\text{1 14}\\ \text{-2 6}\end{array}\right]$

Explanation of Solution

Given:

Matrices are given as;

  $A=\left[\begin{array}{l}1\text{ 3}\\ \text{4 1}\end{array}\right],B=\left[\begin{array}{l}5\text{ -1}\\ \text{-2 0}\end{array}\right]$

Dimension of matrix A is $2\times 2$ and size of matrix B is $2\times 2$ as the number of column of matrix A is equal to number of row of matrix B therefore product AB is possible.

And size of matrix BA will be $2\times 2$

Therefore. product of matrix

  $\begin{array}{l}BA=\left[\begin{array}{l}5\text{ -1}\\ \text{-2 0}\end{array}\right]\left[\begin{array}{l}1\text{ 3}\\ \text{4 1}\end{array}\right]\\ =\left[\begin{array}{l}1\times 5-1\times 4\text{ 5}\times \text{3-1}\times 1\\ -2\times 1-0\times 4\text{ -2}\times \text{3-0}\times 1\end{array}\right]\\ =\left[\begin{array}{l}\text{1 14}\\ \text{-2 6}\end{array}\right]\end{array}$

c.

To determine

Square of the matrix.

Answer to Problem 57RE

Square of the matrix $A^2$ is;

  $\left[\begin{array}{l}13\text{ 6}\\ \text{8 13}\end{array}\right]$

Explanation of Solution

Given:

Matrices are given as;

  $A=\left[\begin{array}{l}1\text{ 3}\\ \text{4 1}\end{array}\right],B=\left[\begin{array}{l}5\text{ -1}\\ \text{-2 0}\end{array}\right]$

Dimension of matrix A is $2\times 2$ so it is a square matrix

Therefore square of the matrix $A^2$ is possible.

  $\begin{array}{l}{A}^2=\left[\begin{array}{l}1\text{ 3}\\ \text{4 1}\end{array}\right]\left[\begin{array}{l}1\text{ 3}\\ \text{4 1}\end{array}\right]\\ =\left[\begin{array}{l}1\times 1+3\times 4\text{ 1}\times \text{3+3}\times \text{1}\\ 4\times 1+1\times 4\text{ 4}\times \text{3+1}\times \text{1}\end{array}\right]\\ =\left[\begin{array}{l}13\text{ 6}\\ \text{8 13}\end{array}\right]\end{array}$