Question

Use the following matrices to evaluate the given expression.

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

$AC$

Short answer

$AC=\left[\begin{array}{cc}28& -9\\ 4& 23\end{array}\right]$

Step-by-step solution

Step 1. Given Information

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

Step 2. Calculation

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