Chapter 7.2, Problem 21E

Solution

a.

To calculate: The matrix multiplication $AB$ 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 $\left[\begin{array}{ccc}6& -7& -2\\ 3& 7& 3\\ 8& -1& -1\end{array}\right]$ .

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 $A$ is equal to the number of row of the matrix $B$ .

Now, find the value of $AB$ .

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

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

  

Hence, the obtained matrix after multiplication is $\left[\begin{array}{ccc}6& -7& -2\\ 3& 7& 3\\ 8& -1& -1\end{array}\right]$ .

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 $\left[\begin{array}{ccc}2& 1& 3\\ 5& 0& 0\\ -18& -3& 10\end{array}\right]$ .

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 $BA$ .

  $\begin{array}{c}BA=\left[\begin{array}{ccc}2& 1& 0\\ -1& 0& 2\\ 4& -3& -1\end{array}\right]\cdot \left[\begin{array}{ccc}-1& 0& 2\\ 4& 1& -1\\ 2& 0& 1\end{array}\right]\\ =\left[\begin{array}{ccc}-2+4+0& 0+1+0& 4-1+0\\ 1+0+4& 0+0+0& -2+0+2\\ -4-12-2& 0-3-0& 8+3-1\end{array}\right]\\ =\left[\begin{array}{ccc}2& 1& 3\\ 5& 0& 0\\ -18& -3& 10\end{array}\right]\end{array}$

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

  

Hence, the obtained matrix after multiplication is $\left[\begin{array}{ccc}2& 1& 3\\ 5& 0& 0\\ -18& -3& 10\end{array}\right]$ .