Chapter 7.2, Problem 25E

Solution

a.

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

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

The multiplication does not exists.

Given Information:

The matrix are,

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

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,

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

It can be observed that the number of column of matrix $A$ is 2 and the number of row of the matrix $B$ is 1.

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

Hence, the multiplication does not exists.

b.

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

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

The obtained matrix is $\left[\begin{array}{cc}18& 14\end{array}\right]$ .

Given Information:

The matrix are,

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

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,

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

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

Now, find the value of $BA$ .

  $\begin{array}{c}BA=\left[\begin{array}{cc}-3& 5\end{array}\right]\cdot \left[\begin{array}{cc}-1& 2\\ 3& 4\end{array}\right]\\ =\left[\begin{array}{cc}-3\left(-1\right)+5\left(3\right)& -3\left(2\right)+5\left(4\right)\end{array}\right]\\ =\left[\begin{array}{cc}3+15& -6+20\end{array}\right]\\ =\left[\begin{array}{cc}18& 14\end{array}\right]\end{array}$

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

  

Hence, the obtained matrix after multiplication is $\left[\begin{array}{cc}18& 14\end{array}\right]$ .