Q. 21 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 9–24, use the following matrices to evaluate the given expression.

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

$AC-3I_2$

Step-by-Step Solution

Answer

The evaluation of the expression $AC-3I_2$ is $\left[\begin{array}{cc}25& -9\\ 4& 20\end{array}\right]$

Step 1. Given Information

The matrices:

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

Step 2. Find AC

$$\begin{gathered} AC=\left[\begin{array}{ccc}0& 3& -5\\ 1& 2& 6\end{array}\right]\left[\begin{array}{cc}4& 1\\ 6& 2\\ -2& 3\end{array}\right] \\ =\left[\begin{array}{cc}0\left(4\right)+3\left(6\right)+-5(-2)& 0\left(1\right)+3\left(2\right)+-5\left(3\right)\\ 1\left(4\right)+2\left(6\right)+6(-2)& 1\left(1\right)+2\left(2\right)+6\left(3\right)\end{array}\right] \\ =\left[\begin{array}{cc}28& -9\\ 4& 23\end{array}\right] \end{gathered}$$

Step 3. Find 3 I 2

$$\begin{gathered} 3I_2=3\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right] \\ =\left[\begin{array}{cc}3& 0\\ 0& 3\end{array}\right] \end{gathered}$$

Step 4. Find the resultant matrix

$$\begin{gathered} AC-3I_2=\left[\begin{array}{cc}28& -9\\ 4& 23\end{array}\right]-\left[\begin{array}{cc}3& 0\\ 0& 3\end{array}\right] \\ =\left[\begin{array}{cc}25& -9\\ 4& 20\end{array}\right] \end{gathered}$$