Chapter 7.2, Problem 22E

Solution

a.

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

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

The obtained matrix is $\left[\begin{array}{ccc}-8& 8& 5\\ 0& 7& -8\\ 11& 4& 11\end{array}\right]$ .

Given Information:

The matrix are,

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

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,

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

It can be observed that the number of column of matrix $A$ is equal to the number of row of the matrix $B$ .

Now, find the value of $AB$ .

  $\begin{array}{c}AB=\left[\begin{array}{ccc}-2& 3& 0\\ 1& -2& 4\\ 3& 2& 1\end{array}\right]\cdot \left[\begin{array}{ccc}4& -1& 2\\ 0& 2& 3\\ -1& 3& -1\end{array}\right]\\ =\left[\begin{array}{ccc}-8+0+0& 2+6+0& -4+9+0\\ 4+0-4& -1-4+12& 2-6-4\\ 12+0-1& -3+4+3& 6+6-1\end{array}\right]\\ =\left[\begin{array}{ccc}-8& 8& 5\\ 0& 7& -8\\ 11& 4& 11\end{array}\right]\end{array}$

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

  

Hence, the obtained matrix after multiplication is $\left[\begin{array}{ccc}-8& 8& 5\\ 0& 7& -8\\ 11& 4& 11\end{array}\right]$ .

b.

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

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

The obtained matrix is $\left[\begin{array}{ccc}-3& 18& -2\\ 11& 2& 11\\ 2& -11& 11\end{array}\right]$ .

Given Information:

The matrix are,

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

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,

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

It can be observed that the number of column of matrix $B$ is equal to the number of row of the matrix $A$ .

Now, find the value of $BA$ .

  $\begin{array}{c}BA=\left[\begin{array}{ccc}2& 1& 0\\ -1& 0& 2\\ 4& -3& -1\end{array}\right]\cdot \left[\begin{array}{ccc}-1& 0& 2\\ 4& 1& -1\\ 2& 0& 1\end{array}\right]\\ =\left[\begin{array}{ccc}-8-1+6& 12+2+4& 0-4+2\\ 0+2+9& 0-4+6& 0+8+3\\ 2+3-3& -3-6-2& 0+12-1\end{array}\right]\\ =\left[\begin{array}{ccc}-3& 18& -2\\ 11& 2& 11\\ 2& -11& 11\end{array}\right]\end{array}$

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

  

Hence, the obtained matrix after multiplication is $\left[\begin{array}{ccc}-3& 18& -2\\ 11& 2& 11\\ 2& -11& 11\end{array}\right]$ .