Chapter 7.2, Problem 26E

Solution

a.

To calculate: The matrix multiplication $AB$ by using the definition of the matrix multiplication.

  $A=\left[\begin{array}{cc}-1& 3\\ 0& 1\\ 1& 0\\ -3& -1\end{array}\right]$ , $B=\left[\begin{array}{cc}5& -6\\ 2& 3\end{array}\right]$

The multiplication of matrix is $\left[\begin{array}{cc}1& 15\\ 2& 3\\ 5& -6\\ -17& 15\end{array}\right]$ .

Given Information:

The matrix are,

  $A=\left[\begin{array}{cc}-1& 3\\ 0& 1\\ 1& 0\\ -3& -1\end{array}\right]$ , $B=\left[\begin{array}{cc}5& -6\\ 2& 3\end{array}\right]$

Formula Used:

The matrix multiplication is possible if and only if order of the first matrix column equal to the order of the second matrix row.

Calculation:

Consider the given matrix,

  $A=\left[\begin{array}{cc}-1& 3\\ 0& 1\\ 1& 0\\ -3& -1\end{array}\right]$ , $B=\left[\begin{array}{cc}5& -6\\ 2& 3\end{array}\right]$

It can be observed that the number of column of matrix $A$ is 2 and the number of row of the matrix $B$ is 2. So the multiplication of the matrix $AB$ exists.

Now, find the value of $AB$ .

  $\begin{array}{c}AB=\left[\begin{array}{cc}-1& 3\\ 0& 1\\ 1& 0\\ -3& -1\end{array}\right]\cdot \left[\begin{array}{cc}5& -6\\ 2& 3\end{array}\right]\\ =\left[\begin{array}{cc}-5+6& 6+9\\ 0+2& 0+3\\ 5+0& -6+0\\ -15-2& 18-3\end{array}\right]\\ =\left[\begin{array}{cc}1& 15\\ 2& 3\\ 5& -6\\ -17& 15\end{array}\right]\end{array}$

Now, use the graphing calculator to find the product of the matrix.

  

Hence, the obtained matrix after multiplication is $\left[\begin{array}{cc}1& 15\\ 2& 3\\ 5& -6\\ -17& 15\end{array}\right]$ .

b.

To calculate: The matrix multiplication $BA$ by using the definition of the matrix multiplication.

  $A=\left[\begin{array}{cc}-1& 3\\ 0& 1\\ 1& 0\\ -3& -1\end{array}\right]$ , $B=\left[\begin{array}{cc}5& -6\\ 2& 3\end{array}\right]$

The multiplication does not exists.

Given Information:

The matrix are,

  $A=\left[\begin{array}{cc}-1& 3\\ 0& 1\\ 1& 0\\ -3& -1\end{array}\right]$ , $B=\left[\begin{array}{cc}5& -6\\ 2& 3\end{array}\right]$

Formula Used:

The matrix multiplication is possible if and only if order of the first matrix column equal to the order of the second matrix row.

Calculation:

Consider the given matrix,

  $A=\left[\begin{array}{cc}-1& 3\\ 0& 1\\ 1& 0\\ -3& -1\end{array}\right]$ , $B=\left[\begin{array}{cc}5& -6\\ 2& 3\end{array}\right]$

It can be observed that the number of column of matrix $B$ is 2 and the number of row of the matrix $A$ is 4.

The product of the given matrix $BA$ does not exists.

Hence, the multiplication does not exists.