College Algebra, Chapter 7, 7.2, Section 7.2, Problem 22

Suppose the matrices $A, B, C, D, E, F, G$ and $H$ are defined as



$
\begin{equation}
\begin{aligned}


A =& \left[ \begin{array}{cc}
2 & -5 \\
0 & 7
\end{array}
\right]

&& B = \left[ \begin{array}{ccc}
3 & \displaystyle \frac{1}{2} & 5 \\
1 & -1 & 3
\end{array} \right]

&&& C = \left[ \begin{array}{ccc}
2 & \displaystyle \frac{-5}{2} & 0 \\
0 & 2 & -3
\end{array} \right]

&&&& D = \left[ \begin{array}{cc}
7 & 3
\end{array} \right]
\\
\\
\\
\\
E =& \left[ \begin{array}{c}
1 \\
2 \\
0
\end{array}
\right]

&& F = \left[ \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}
\right]

&&& G = \left[ \begin{array}{ccc}
5 & -3 & 10 \\
6 & 1 & 0 \\
-5 & 2 & 2
\end{array} \right]

&&&& H = \left[ \begin{array}{cc}
3 & 1 \\
2 & -1
\end{array} \right]


\end{aligned}
\end{equation}
$


Carry out the indicated algebraic operation, or explain why it cannot be performed.

a.) $C - B$


$
\begin{equation}
\begin{aligned}

\left[ \begin{array}{ccc}
2 & \displaystyle \frac{-5}{2} & 0 \\
0 & 2 & -3
\end{array} \right]
-
\left[ \begin{array}{ccc}
3 & \displaystyle \frac{1}{2} & 5 \\
1 & -1 & 3
\end{array} \right]

=&

\left[ \begin{array}{ccc}
2-3 & \displaystyle \frac{-5}{2} - \frac{1}{2} & 0-5 \\
0-1 & 2-(-1) & -3-3
\end{array} \right]

&& \text{Substitute the matrices $C$ and $B$}
\\
\\
\\
\\

=& \left[ \begin{array}{ccc}
-1 & -3 & -5 \\
-1 & 3 & -6
\end{array} \right]
&& \text{Simplify}

\end{aligned}
\end{equation}
$


b.) $2C - 6B$


$
\begin{equation}
\begin{aligned}

2 \left[ \begin{array}{ccc}
2 & \displaystyle \frac{-5}{2} & 0 \\
0 & 2 & -3
\end{array} \right] -

6 \left[ \begin{array}{ccc}
3 & \displaystyle \frac{1}{2} & 5 \\
1 & -1 & 3
\end{array} \right]

=& \left[ \begin{array}{ccc}
4 & -5 & 0 \\
0 & 4 & -6
\end{array} \right]

-

\left[ \begin{array}{ccc}
18 & 3 & 30 \\
6 & -6 & 18
\end{array} \right]

\\
\\
\\
\\

=& \left[ \begin{array}{ccc}
4-18 & -5-3 & 0-30 \\
0-6 & 4-(-6) & -6-18
\end{array} \right]
\\
\\
\\
\\

=& \left[ \begin{array}{ccc}
-14 & -8 & -30 \\
-6 & 10 & -24
\end{array} \right]

\end{aligned}
\end{equation}
$