Q. 27 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 25– 30, find the product.

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

Step-by-Step Solution

Answer

The product of the matrices $\left[\begin{array}{ccc}1& 2& 3\\ 0& -1& 4\end{array}\right]\left[\begin{array}{cc}1& 2\\ -1& 0\\ 2& 4\end{array}\right]$ is $\left[\begin{array}{cc}5& 14\\ 9& 16\end{array}\right]$

Step 1. Given Information

The matrices:

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

Step 2. Observe the matrix

The first matrix is $2\times 3$ matrix and the second matrix is $3\times 2$, so the resultant matrix will be of the form $2\times 2$ matrix

Step 3. Find the product

$$\begin{gathered} \left[\begin{array}{ccc}1& 2& 3\\ 0& -1& 4\end{array}\right]\left[\begin{array}{cc}1& 2\\ -1& 0\\ 2& 4\end{array}\right]=\left[\begin{array}{cc}1\left(1\right)+2(-1)+3\left(2\right)& 1\left(2\right)+2\left(0\right)+3\left(4\right)\\ 0\left(1\right)+-1(-1)+4\left(2\right)& 0\left(2\right)+-1\left(0\right)+4\left(4\right)\end{array}\right] \\ =\left[\begin{array}{cc}5& 14\\ 9& 16\end{array}\right] \end{gathered}$$