Q. 25 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 25– 30, find the product.

$\left[\begin{array}{cc}2& -2\\ 1& 0\end{array}\right]\left[\begin{array}{cccc}2& 1& 4& 6\\ 3& -1& 3& 2\end{array}\right]$

Step-by-Step Solution

Answer

The product of the matrices $\left[\begin{array}{cc}2& -2\\ 1& 0\end{array}\right]\left[\begin{array}{cccc}2& 1& 4& 6\\ 3& -1& 3& 2\end{array}\right]$ is $\left[\begin{array}{cccc}-2& 4& 2& 8\\ 2& 1& 4& 6\end{array}\right]$

Step 1. Given Information

The matrices:

$\left[\begin{array}{cc}2& -2\\ 1& 0\end{array}\right]\left[\begin{array}{cccc}2& 1& 4& 6\\ 3& -1& 3& 2\end{array}\right]$

Step 2. Observe the matrix

The first matrix is $2\times 2$ matrix and the second matrix is $2\times 4$, so the resultant matrix will be of the form $2\times 4$

Step 3. Find the product

$$\begin{gathered} \left[\begin{array}{cc}2& -2\\ 1& 0\end{array}\right]\left[\begin{array}{cccc}2& 1& 4& 6\\ 3& -1& 3& 2\end{array}\right]=\left[\begin{array}{cccc}2\left(2\right)+-2\left(3\right)& 2\left(1\right)+-2(-1)& 2\left(4\right)+-2\left(3\right)& 2\left(6\right)+-2\left(2\right)\\ 1\left(2\right)+0\left(3\right)& 1\left(1\right)+0(-1)& 1\left(4\right)+0\left(3\right)& 1\left(6\right)+0\left(2\right)\end{array}\right] \\ =\left[\begin{array}{cccc}-2& 4& 2& 8\\ 2& 1& 4& 6\end{array}\right] \end{gathered}$$