Question

Use the following matrices to evaluate the given expression.

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

$(A+B)C$

Short answer

$(A+B)C=\left[\begin{array}{cc}50& -3\\ 18& 21\end{array}\right]$

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

$$\begin{gathered} A=\left[\begin{array}{ccc}0& 3& -5\\ 1& 2& 6\end{array}\right],B=\left[\begin{array}{ccc}4& 1& 0\\ -2& 3& -2\end{array}\right],C=\left[\begin{array}{cc}4& 1\\ 6& 2\\ -2& 3\end{array}\right] \\ A+B=\left[\begin{array}{ccc}0& 3& -5\\ 1& 2& 6\end{array}\right]+\left[\begin{array}{ccc}4& 1& 0\\ -2& 3& -2\end{array}\right] \\ =\left[\begin{array}{ccc}4& 4& -5\\ -1& 5& -4\end{array}\right] \\ (A+B)C=\left[\begin{array}{ccc}4& 4& -5\\ -1& 5& -4\end{array}\right]\left[\begin{array}{cc}4& 1\\ 6& 2\\ -2& 3\end{array}\right] \\ =\left[\begin{array}{cc}4\times 4+4\times 6+(-5)(-2)& 4\times 1+4\times 2+(-5)\times 3\\ (-1)\times 4+5\times 6+(-4)(-2)& (-1)\times 1+5\times 2+(-4)3\end{array}\right] \\ =\left[\begin{array}{cc}50& -3\\ 18& 21\end{array}\right] \end{gathered}$$