Chapter 8.2, Problem 50E

(A)

To determine

To find:

The product of two matrices $AB$

Answer to Problem 50E

The product of two matrices $AB={\left[\begin{array}{cc}4& 2\\ -2& 4\end{array}\right]}_{2\times 2}$

Explanation of Solution

Given:

The matrix

  $A={\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right]}_{2\times 2},B={\left[\begin{array}{cc}1& 3\\ -3& 1\end{array}\right]}_{2\times 2}$

Concept used:

Definition of matrix multiplication :-if A $=\left[a_{ij}\right]$ is an $m\times n$ matrix and B $=\left[b_{ij}\right]$ is an $n\times p$ matrix , then the product $AB$ is an $m\times p$ matrix

  $\begin{array}{l}AB=\left[a_{ij}\times b_{ij}\right]\\ AB=\left[c_{ij}\right]\end{array}$

Where $c_{ij}$ = $=a_{i1}{b}_{1j}+a_{i2}{b}_{2j}+a_{i3}{b}_{3j}+\mathrm{..........}+a_{in}{b}_{nj}$

Calculation:

The product $AB$ is an $m\times p$ matrix

  $\begin{array}{l}AB=\left[a_{ij}\times b_{ij}\right]\\ AB=\left[c_{ij}\right]\end{array}$

Where $c_{ij}$ = $=a_{i1}{b}_{1j}+a_{i2}{b}_{2j}+a_{i3}{b}_{3j}+\mathrm{..........}+a_{in}{b}_{nj}$

The matrix

  $A={\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right]}_{2\times 2},B={\left[\begin{array}{cc}1& 3\\ -3& 1\end{array}\right]}_{2\times 2}$

The product of two matrices $AB$

  $\begin{array}{l}A={\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right]}_{2\times 2},B={\left[\begin{array}{cc}1& 3\\ -3& 1\end{array}\right]}_{2\times 2}\\ AB={\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right]}_{2\times 2}{\left[\begin{array}{cc}1& 3\\ -3& 1\end{array}\right]}_{2\times 2}\\ AB={\left[\begin{array}{cc}4& 2\\ -2& 4\end{array}\right]}_{2\times 2}\end{array}$

(B)

To determine

To find:

The product of two matrices $BA$

Answer to Problem 50E

The product of two matrices $BA={\left[\begin{array}{cc}4& 2\\ -2& 4\end{array}\right]}_{2\times 2}$

Explanation of Solution

Given:

The matrix

  $A={\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right]}_{2\times 2},B={\left[\begin{array}{cc}1& 3\\ -3& 1\end{array}\right]}_{2\times 2}$

Concept used:

Definition of matrix multiplication :-if A $=\left[a_{ij}\right]$ is an $m\times n$ matrix and B $=\left[b_{ij}\right]$ is an $n\times p$ matrix , then the product $BA$ is an $m\times p$ matrix

  $\begin{array}{l}BA=\left[b_{ij}\times a_{ij}\right]\\ BA=\left[c_{ij}\right]\end{array}$

Where $c_{ij}$ = $=a_{i1}{b}_{1j}+a_{i2}{b}_{2j}+a_{i3}{b}_{3j}+\mathrm{..........}+a_{in}{b}_{nj}$

Calculation:

The product $AB$ is an $m\times p$ matrix

  $\begin{array}{l}BA=\left[a_{ij}\times b_{ij}\right]\\ BA=\left[c_{ij}\right]\end{array}$

Where $c_{ij}$ = $=a_{i1}{b}_{1j}+a_{i2}{b}_{2j}+a_{i3}{b}_{3j}+\mathrm{..........}+a_{in}{b}_{nj}$

The matrix

  $A={\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right]}_{2\times 2},B={\left[\begin{array}{cc}1& 3\\ -3& 1\end{array}\right]}_{2\times 2}$

The product of two matrices $BA$

  $\begin{array}{l}A={\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right]}_{2\times 2},B={\left[\begin{array}{cc}1& 3\\ -3& 1\end{array}\right]}_{2\times 2}\\ BA={\left[\begin{array}{cc}1& 3\\ -3& 1\end{array}\right]}_{2\times 2}{\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right]}_{2\times 2}\\ BA={\left[\begin{array}{cc}4& 2\\ -2& 4\end{array}\right]}_{2\times 2}\end{array}$

(C)

To determine

To find:

The product of two matrices $A^2$

Answer to Problem 50E

The product of two matrices $A^2={\left[\begin{array}{cc}0& 2\\ 2& 0\end{array}\right]}_{2\times 2}$

Explanation of Solution

Given:

The matrix

  $A={\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right]}_{2\times 2},B={\left[\begin{array}{cc}1& 3\\ -3& 1\end{array}\right]}_{2\times 2}$

Concept used:

Definition of matrix multiplication :-if A $=\left[a_{ij}\right]$ is an $m\times n$ matrix and B $=\left[b_{ij}\right]$ is an $n\times p$ matrix , then the product $AB$ is an $m\times p$ matrix

  $\begin{array}{l}AB=\left[a_{ij}\times b_{ij}\right]\\ AB=\left[c_{ij}\right]\end{array}$

Where $c_{ij}$ = $=a_{i1}{b}_{1j}+a_{i2}{b}_{2j}+a_{i3}{b}_{3j}+\mathrm{..........}+a_{in}{b}_{nj}$

Calculation:

The product $AB$ is an $m\times p$ matrix

  $\begin{array}{l}AB=\left[a_{ij}\times b_{ij}\right]\\ AB=\left[c_{ij}\right]\end{array}$

Where $c_{ij}$ = $=a_{i1}{b}_{1j}+a_{i2}{b}_{2j}+a_{i3}{b}_{3j}+\mathrm{..........}+a_{in}{b}_{nj}$

The matrix

  $A={\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right]}_{2\times 2},B={\left[\begin{array}{cc}1& 3\\ -3& 1\end{array}\right]}_{2\times 2}$

The product of two matrices $AA$

  $\begin{array}{l}A={\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right]}_{2\times 2},B={\left[\begin{array}{cc}1& 3\\ -3& 1\end{array}\right]}_{2\times 2}\\ AA=A^2={\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right]}_{2\times 2}{\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right]}_{2\times 2}\\ A^2={\left[\begin{array}{cc}0& 2\\ 2& 0\end{array}\right]}_{2\times 2}\end{array}$