Q. 28 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 25– 30, find the product.

$\left[\begin{array}{cc}1& -1\\ -3& 2\\ 0& 5\end{array}\right]\left[\begin{array}{ccc}2& 8& -1\\ 3& 6& 0\end{array}\right]$

Step-by-Step Solution

Answer

The product of the matrices $\left[\begin{array}{cc}1& -1\\ -3& 2\\ 0& 5\end{array}\right]\left[\begin{array}{ccc}2& 8& -1\\ 3& 6& 0\end{array}\right]$ is $\left[\begin{array}{ccc}-1& 2& -1\\ 0& -12& 3\\ 15& 30& 0\end{array}\right]$

Step 1. Given Information

The matrices:

$\left[\begin{array}{cc}1& -1\\ -3& 2\\ 0& 5\end{array}\right]\left[\begin{array}{ccc}2& 8& -1\\ 3& 6& 0\end{array}\right]$

Step 2. Observe the matrix

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

Step 3. Find the product

$$\begin{gathered} \left[\begin{array}{cc}1& -1\\ -3& 2\\ 0& 5\end{array}\right]\left[\begin{array}{ccc}2& 8& -1\\ 3& 6& 0\end{array}\right]=\left[\begin{array}{ccc}1\left(2\right)+-1\left(3\right)& 1\left(8\right)+-1\left(6\right)& 1(-1)+-1\left(0\right)\\ -3\left(2\right)+2\left(3\right)& -3\left(8\right)+2\left(6\right)& -3(-1)+2\left(0\right)\\ 0\left(2\right)+5\left(3\right)& 0\left(8\right)+5\left(6\right)& 0(-1)+5\left(0\right)\end{array}\right] \\ =\left[\begin{array}{ccc}-1& 2& -1\\ 0& -12& 3\\ 15& 30& 0\end{array}\right] \end{gathered}$$