Chapter 7.2, Problem 23E

Solution

a.

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

  $A=\left[\begin{array}{ccc}2& -1& 3\end{array}\right]$ , $B=\left[\begin{array}{c}-5\\ 4\\ 2\end{array}\right]$

The obtained matrix is $\left[-8\right]$ .

Given Information:

The matrix are,

  $A=\left[\begin{array}{ccc}2& -1& 3\end{array}\right]$ , $B=\left[\begin{array}{c}-5\\ 4\\ 2\end{array}\right]$

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,

  $A=\left[\begin{array}{ccc}2& -1& 3\end{array}\right]$ , $B=\left[\begin{array}{c}-5\\ 4\\ 2\end{array}\right]$

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 $AB$ .

  $\begin{array}{c}AB=\left[\begin{array}{ccc}2& -1& 3\end{array}\right]\cdot \left[\begin{array}{c}-5\\ 4\\ 2\end{array}\right]\\ =\left[2\left(-5\right)+\left(-1\right)4+3\left(2\right)\right]\\ =\left[-10-4+6\right]\\ =\left[-8\right]\end{array}$

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

  

Hence, the obtained matrix after multiplication is $\left[-8\right]$ .

b.

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

  $A=\left[\begin{array}{ccc}2& -1& 3\end{array}\right]$ , $B=\left[\begin{array}{c}-5\\ 4\\ 2\end{array}\right]$

The obtained matrix is $\left[\begin{array}{ccc}-10& 5& -15\\ 8& -4& 12\\ 4& -2& 6\end{array}\right]$ .

Given Information:

The matrix are,

  $A=\left[\begin{array}{ccc}2& -1& 3\end{array}\right]$ , $B=\left[\begin{array}{c}-5\\ 4\\ 2\end{array}\right]$

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,

  $A=\left[\begin{array}{ccc}2& -1& 3\end{array}\right]$ , $B=\left[\begin{array}{c}-5\\ 4\\ 2\end{array}\right]$

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 $BA$ .

  $\begin{array}{c}BA=\left[\begin{array}{c}-5\\ 4\\ 2\end{array}\right]\cdot \left[\begin{array}{ccc}2& -1& 3\end{array}\right]\\ =\left[\begin{array}{ccc}-5\left(2\right)& -5\left(-1\right)& -5\left(3\right)\\ 4\left(2\right)& 4\left(-1\right)& 4\left(3\right)\\ 2\left(2\right)& 2\left(-1\right)& 2\left(3\right)\end{array}\right]\\ =\left[\begin{array}{ccc}-10& 5& -15\\ 8& -4& 12\\ 4& -2& 6\end{array}\right]\end{array}$

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

  

Hence, the obtained matrix after multiplication is $\left[\begin{array}{ccc}-10& 5& -15\\ 8& -4& 12\\ 4& -2& 6\end{array}\right]$ .