Question

In Problems 9–24, use the following matrices to evaluate the given expression.

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

$AC+BC$

Short answer

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

Step-by-step solution

Step 1. Given Information

The matrices:

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

Step 2. Find AC 

$$\begin{gathered} AC=\left[\begin{array}{ccc}0& 3& -5\\ 1& 2& 6\end{array}\right]\left[\begin{array}{cc}4& 1\\ 6& 2\\ -2& 3\end{array}\right] \\ =\left[\begin{array}{cc}0\left(4\right)+3\left(6\right)+-5(-2)& 0\left(1\right)+3\left(2\right)+-5\left(3\right)\\ 1\left(4\right)+2\left(6\right)+6(-2)& 1\left(1\right)+2\left(2\right)+6\left(3\right)\end{array}\right] \\ =\left[\begin{array}{cc}28& -9\\ 4& 23\end{array}\right] \end{gathered}$$

Step 3. Find BC

$$\begin{gathered} BC=\left[\begin{array}{ccc}4& 1& 0\\ -2& 3& -2\end{array}\right]\left[\begin{array}{cc}4& 1\\ 6& 2\\ -2& 3\end{array}\right] \\ =\left[\begin{array}{cc}4\left(4\right)+1\left(6\right)+0(-2)& 4\left(1\right)+1\left(2\right)+0\left(3\right)\\ -2\left(4\right)+3\left(6\right)+-2(-2)& -2\left(1\right)+3\left(2\right)+-2\left(3\right)\end{array}\right] \\ =\left[\begin{array}{cc}22& 6\\ 14& -2\end{array}\right] \end{gathered}$$

Step 4. Find the resultant matrix

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