Chapter 7.5, Problem 48E

(a).

To determine

To calculate: the product of two matrices.

Answer to Problem 48E

The product of two matrix $A$ and $B$ that is $\underset{\_}{AB=\left[\begin{array}{cc}4& 2\\ -2& 4\end{array}\right]}$ .

Explanation of Solution

Given information:

Two matrix for which product that is $AB$ needs to be calculated are as follows.

  $A=\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right]$ and $B=\left[\begin{array}{cc}1& 3\\ -3& 1\end{array}\right]$ ..

Formula used:

To calculate product of two matrix , prerequisite is number of columns in first matrix must be equal to number of rows in second matrix.

Calculation:

Two matrix for which product $AB$ needs to be calculated are as follows.

  $A=\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right]$ and $B=\left[\begin{array}{cc}1& 3\\ -3& 1\end{array}\right]$ .

As number of columns of matrix $A$ is equal to number of rows of matrix $B$ that is 2 , so $AB$ can be calculated.

Calculate product

  $AB=\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right]\left[\begin{array}{cc}1& 3\\ -3& 1\end{array}\right]$

  $=\left[\begin{array}{cc}1\times 1+(-1)\times (-3)& 1\times 3+(-1)\times 1\\ 1\times 1+1\times (-3)& 1\times 3+1\times 1\end{array}\right]$

  $=\left[\begin{array}{cc}1+3& 3-1\\ 1-3& 3+1\end{array}\right]$

  $=\left[\begin{array}{cc}4& 2\\ -2& 4\end{array}\right]$

Hence the product of two matrix $A$ and $B$ that is $\underset{\_}{AB=\left[\begin{array}{cc}4& 2\\ -2& 4\end{array}\right]}$ .

(b).

To determine

To calculate: the product of two matrices.

Answer to Problem 48E

The product of two matrix $B$ and $A$ that is $\underset{\_}{AB=\left[\begin{array}{cc}4& 2\\ -2& 4\end{array}\right]}$ .

Explanation of Solution

Given information:

Two matrix for which product that is $BA$ needs to be calculated are as follows.

  $A=\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right]$ and $B=\left[\begin{array}{cc}1& 3\\ -3& 1\end{array}\right]$ ..

Formula used:

To calculate product of two matrix , prerequisite is number of columns in first matrix must be equal to number of rows in second matrix.

Calculation:

Two matrix for which product $BA$ needs to be calculated are as follows.

  $A=\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right]$ and $B=\left[\begin{array}{cc}1& 3\\ -3& 1\end{array}\right]$

As number of columns of matrix $B$ is equal to number of number of rows of matrix $A$ that is 2 , so $BA$ can be calculated.

Calculate product

  $BA=\left[\begin{array}{cc}1& 3\\ -3& 1\end{array}\right]\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right]$

  $=\left[\begin{array}{cc}1\times 1+3\times 1& 1\times (-1)+3\times 1\\ (-3)\times 1+1\times 1& (-3)\times (-1)+1\times 1\end{array}\right]$

  $=\left[\begin{array}{cc}1+3& -1+3\\ -3+1& 3+1\end{array}\right]$

  $=\left[\begin{array}{cc}4& 2\\ -2& 4\end{array}\right]$

Hence the product of two matrix $B$ and $A$ that is $\underset{\_}{BA=\left[\begin{array}{cc}0& -10\\ 10& 0\end{array}\right]}$ .

(c).

To determine

To calculate: the square of a matrix.

Answer to Problem 48E

The square of matrix $A$ that is $\underset{\_}{A^2=\left[\begin{array}{cc}0& -2\\ 2& 0\end{array}\right]}$

Explanation of Solution

Given information:

The matrix for which the square that is $A^2=A.A$ needs to be calculated is as follows.

  $A=\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right]$ ..

Formula used:

To calculate product of two matrix, prerequisite is number of columns in first matrix must be equal to number of rows in second matrix.

Calculation:

The matrix for which square that is $A^2=A.A$ needs to be calculated is as follows.

  $A=\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right]$

Calculate the square

  $A^2=A.A$

  $=\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right]\left[\begin{array}{cc}1& -1\\ 1& 1\end{array}\right]$

  $=\left[\begin{array}{cc}1\times 1+(-1)\times 1& 1\times (-1)+(-1)\times 1\\ 1\times 1+1\times 1& 1\times (-1)+1\times 1\end{array}\right]$

  $=\left[\begin{array}{cc}1-1& -1-1\\ 1+1& -1+1\end{array}\right]$

  $=\left[\begin{array}{cc}0& -2\\ 2& 0\end{array}\right]$

Hence the square of matrix $A$ that is $\underset{\_}{A^2=\left[\begin{array}{cc}0& -2\\ 2& 0\end{array}\right]}$ .