Chapter 11.4, Problem 24AYU

To determine

To calculate: The value of the expression $AC+BC$ .

Answer to Problem 24AYU

The value of the expression $AC+BC$ 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 matrix B.

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, $AC$ . 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 matrix B.

  $\begin{array}{c}AC=\left[\begin{array}{ccc}0& 3& -5\\ 1& 2& 6\end{array}\right]\left[\begin{array}{cc}4& 1\\ 6& 2\\ -2& 3\end{array}\right]\\ =\left[\begin{array}{cc}0+18+10& 0+6-15\\ 4+12-12& 1+4+18\end{array}\right]\\ =\left[\begin{array}{cc}28& -9\\ 4& 23\end{array}\right]\end{array}$

Now, compute $BC$ .

  $\begin{array}{c}BC=\left[\begin{array}{ccc}4& 1& 0\\ -2& 3& -2\end{array}\right]\left[\begin{array}{cc}4& 1\\ 6& 2\\ -2& 3\end{array}\right]\\ =\left[\begin{array}{cc}16+6+0& 4+2+0\\ -8+18+4& -2+6-6\end{array}\right]\\ =\left[\begin{array}{cc}22& 6\\ 14& -2\end{array}\right]\end{array}$

Next, compute $AC+BC$ ,

  $\begin{array}{c}AC+BC=\left[\begin{array}{cc}28& -9\\ 4& 23\end{array}\right]+\left[\begin{array}{cc}22& 6\\ 14& -2\end{array}\right]\\ =\left[\begin{array}{cc}50& -3\\ 18& 21\end{array}\right]\end{array}$

Thus, the value of the expression $AC+BC$ is $\left[\begin{array}{cc}50& -3\\ 18& 21\end{array}\right]$ .