Chapter 11.4, Problem 20AYU

To determine

To calculate: The value of the expression $\left(A+B\right)C$ .

Answer to Problem 20AYU

The value of the expression $\left(A+B\right)C$ is $\left[\begin{array}{cc}50& -3\\ 18& 21\end{array}\right]$ .

Explanation of Solution

Given information:

The matrices are $A=\left[\begin{array}{ccc}0& 3& -5\\ 1& 2& 6\end{array}\right],B=\left[\begin{array}{ccc}4& 1& 0\\ -2& 3& -2\end{array}\right],C=\left[\begin{array}{cc}4& 1\\ 6& 2\\ -2& 3\end{array}\right]$

Formula used:

To find the product AB of two matrices A and B , the number of column in matrix A must be equal to number of rows in matrixB .

Calculation:

Consider the matrices are $A=\left[\begin{array}{ccc}0& 3& -5\\ 1& 2& 6\end{array}\right],B=\left[\begin{array}{ccc}4& 1& 0\\ -2& 3& -2\end{array}\right],C=\left[\begin{array}{cc}4& 1\\ 6& 2\\ -2& 3\end{array}\right]$

First compute, $A+B$ , add the corresponding row and column elements.

  $\begin{array}{c}A+B=\left[\begin{array}{ccc}0& 3& -5\\ 1& 2& 6\end{array}\right]+\left[\begin{array}{ccc}4& 1& 0\\ -2& 3& -2\end{array}\right]\\ =\left[\begin{array}{ccc}4& 4& -5\\ -1& 5& 4\end{array}\right]\end{array}$

Now, compute $\left(A+B\right)C$ .

Recall that to find the product AB of two matrices A and B , the number of column in matrix A must be equal to number of rows in matrixB .

  $\begin{array}{c}\left(A+B\right)C=\left[\begin{array}{ccc}4& 4& -5\\ -1& 5& 4\end{array}\right]\left[\begin{array}{cc}4& 1\\ 6& 2\\ -2& 3\end{array}\right]\\ =\left[\begin{array}{cc}16+24+10& 4+8-15\\ -4+30-8& -1+10+12\end{array}\right]\\ =\left[\begin{array}{cc}50& -3\\ 18& 21\end{array}\right]\end{array}$

Thus, the value of the expression $\left(A+B\right)C$ is $\left[\begin{array}{cc}50& -3\\ 18& 21\end{array}\right]$ .