Question

In Problems 13–16, use the following matrices to compute each expression.

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

$AB$

Short answer

$AB=\left[\begin{array}{ccc}4& -3& 0\\ 12& -2& -8\\ -2& 5& -4\end{array}\right]$

Step-by-step solution

Step 1. Given information

The given matrices are

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

Step 2. Find the value of expression AB.

$$\begin{gathered} AB \\ =\left[\begin{array}{cc}1& 0\\ 2& 4\\ -1& 2\end{array}\right]\left[\begin{array}{ccc}4& -3& 0\\ 1& 1& -2\end{array}\right] \\ =\left[\begin{array}{ccc}1·4+0·1& 1·\left(-3\right)+0·1& 1·0+0·\left(-2\right)\\ 2·4+4·1& 2·\left(-3\right)+4·1& 2·0+4·\left(-2\right)\\ \left(-1\right)·4+2·1& \left(-1\right)·\left(-3\right)+2·1& \left(-1\right)·0+2·\left(-2\right)\end{array}\right] \\ =\left[\begin{array}{ccc}4& -3& 0\\ 12& -2& -8\\ -2& 5& -4\end{array}\right] \end{gathered}$$