Chapter 2.3, Problem 45E

To determine

To find: value for the given case by using the matrices $A,B,C,D,E$ and $F$ . Write impossible if the matrix does not exist.

Answer to Problem 45E

  $\left[\begin{array}{ccc}\begin{array}{l}-42\\ -421\end{array}& \begin{array}{l}86\\ 213\end{array}& \begin{array}{l}-160\\ -111\end{array}\end{array}\right]$

Explanation of Solution

Given information:

The given matrices are as follows:

  $\begin{array}{ccc}\begin{array}{l}A=\left[\begin{array}{cc}5& 7\\ -6& 1\end{array}\right]\\ B=\left[\begin{array}{cc}3& 5\\ -1& 8\end{array}\right]\end{array}& \begin{array}{l}C=\left[\begin{array}{ccc}\begin{array}{l}4\\ 5\\ 9\end{array}& \begin{array}{l}-2\\ 0\\ 0\end{array}& \begin{array}{l}3\\ -1\\ 1\end{array}\end{array}\right]\\ D=\left[\begin{array}{ccc}\begin{array}{l}0\\ -2\\ 4\end{array}& \begin{array}{l}1\\ 3\\ 4\end{array}& \begin{array}{l}2\\ 0\\ -2\end{array}\end{array}\right]\end{array}& \begin{array}{l}E=\left[\begin{array}{ccc}\begin{array}{l}8\\ 3\end{array}& \begin{array}{l}-4\\ 1\end{array}& \begin{array}{l}2\\ -5\end{array}\end{array}\right]\\ F=\left[\begin{array}{ccc}\begin{array}{l}-6\\ 1\end{array}& \begin{array}{l}-1\\ 4\end{array}& \begin{array}{l}0\\ 0\end{array}\end{array}\right]\end{array}\end{array}$

The given case is $\left(BA\right)E$ .

Calculation:

First solve $BA$ by taking the sum of the products of the corresponding elements in the $i^{\text{th}}$ row of $B$ (first matrix) and the $j^{\text{th}}$ column of $A$ (second matrix), as shown below.

  $\begin{array}{c}BA=\left[\begin{array}{cc}3& 5\\ -1& 8\end{array}\right]\left[\begin{array}{cc}5& 7\\ -6& 1\end{array}\right]\\ =\left[\begin{array}{cc}15-30& 21+5\\ -5-48& -7+8\end{array}\right]\\ =\left[\begin{array}{cc}-15& 26\\ -53& 1\end{array}\right]\end{array}$

Now, multiply $BA$ by $E$ as shown below.

  $\begin{array}{c}\left(BA\right)E=\left[\begin{array}{cc}-15& 26\\ -53& 1\end{array}\right]\left[\begin{array}{ccc}\begin{array}{l}8\\ 3\end{array}& \begin{array}{l}-4\\ 1\end{array}& \begin{array}{l}2\\ -5\end{array}\end{array}\right]\\ =\left[\begin{array}{ccc}\begin{array}{l}-120+78\\ -424+3\end{array}& \begin{array}{l}60+26\\ 212+1\end{array}& \begin{array}{l}-30-130\\ -106-5\end{array}\end{array}\right]\\ =\left[\begin{array}{ccc}\begin{array}{l}-42\\ -421\end{array}& \begin{array}{l}86\\ 213\end{array}& \begin{array}{l}-160\\ -111\end{array}\end{array}\right]\end{array}$

Thus, the values of $\left(BA\right)E$ is $\left[\begin{array}{ccc}\begin{array}{l}-42\\ -421\end{array}& \begin{array}{l}86\\ 213\end{array}& \begin{array}{l}-160\\ -111\end{array}\end{array}\right]$ .