Question

Use the following matrices to evaluate the given expression.

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

Short answer

$C(A+B)=\left[\begin{array}{ccc}15& 21& -16\\ 22& 34& -22\\ -11& 7& 22\end{array}\right]$

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

$$\begin{gathered} A=\left[\begin{array}{ccc}0& 3& -5\\ 1& 2& 6\end{array}\right],B=\left[\begin{array}{ccc}4& 1& 0\\ -2& 3& -2\end{array}\right],C=\left[\begin{array}{cc}4& 1\\ 6& 2\\ -2& 3\end{array}\right] \\ A+B=\left[\begin{array}{ccc}0& 3& -5\\ 1& 2& 6\end{array}\right]+\left[\begin{array}{ccc}4& 1& 0\\ -2& 3& -2\end{array}\right] \\ =\left[\begin{array}{ccc}4& 4& -5\\ -1& 5& 4\end{array}\right] \\ C(A+B)=\left[\begin{array}{cc}4& 1\\ 6& 2\\ -2& 3\end{array}\right]\left[\begin{array}{ccc}4& 4& -5\\ -1& 5& 4\end{array}\right] \\ =\left[\begin{array}{ccc}4\times 4+6\times (-1)& 4\times 4+1\times 5& 4(-5)+1\times 4\\ 6\times 4+2(-1)& 6\times 4+2\times 5& 6(-5)+2\times 4\\ (-2)\times 4+3(-1)& (-2)\times 4+3\times 5& (-2)(-5)+3\times 4\end{array}\right] \\ =\left[\begin{array}{ccc}15& 21& -16\\ 22& 34& -22\\ -11& 7& 22\end{array}\right] \end{gathered}$$