Q. 22 - Precalculus Enhanced with Graphing Utilities

Question

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

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

$CA+5I_3$

Step-by-Step Solution

Answer

The evaluation of the expression $CA+5I_3$ is $\left[\begin{array}{ccc}6& 14& -14\\ 2& 27& -18\\ 3& 0& 33\end{array}\right]$

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 CA

$$\begin{gathered} CA=\left[\begin{array}{cc}4& 1\\ 6& 2\\ -2& 3\end{array}\right]\left[\begin{array}{ccc}0& 3& -5\\ 1& 2& 6\end{array}\right] \\ =\left[\begin{array}{ccc}4\left(0\right)+1\left(1\right)& 4\left(3\right)+1\left(2\right)& 4(-5)+1\left(6\right)\\ 6\left(0\right)+2\left(1\right)& 6\left(3\right)+2\left(2\right)& 6(-5)+2\left(6\right)\\ -2\left(0\right)+3\left(1\right)& -2\left(3\right)+3\left(2\right)& -2(-5)+3\left(6\right)\end{array}\right] \\ =\left[\begin{array}{ccc}1& 14& -14\\ 2& 22& -18\\ 3& 0& 28\end{array}\right] \end{gathered}$$

Step 3. Find 5 I 3

$$\begin{gathered} 5I_3=5\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right] \\ =\left[\begin{array}{ccc}5& 0& 0\\ 0& 5& 0\\ 0& 0& 5\end{array}\right] \end{gathered}$$

Step 4. Find the resultant matrix

$$\begin{gathered} CA+5I_3=\left[\begin{array}{ccc}1& 14& -14\\ 2& 22& -18\\ 3& 0& 28\end{array}\right]+\left[\begin{array}{ccc}5& 0& 0\\ 0& 5& 0\\ 0& 0& 5\end{array}\right] \\ =\left[\begin{array}{ccc}6& 14& -14\\ 2& 27& -18\\ 3& 0& 33\end{array}\right] \end{gathered}$$