College Algebra, Chapter 7, 7.2, Section 7.2, Problem 30

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.) $B^2$


$
\begin{equation}
\begin{aligned}

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

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

\end{aligned}
\end{equation}
$


But the product of the matrices $B$ and $B$ can not be solve because the number of columns of the first matrix must equal the number of rows of the second matrix.

b.) $F^2$

$\displaystyle F^2 = \left[ \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right]
\left[ \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right]
$


$
\begin{equation}
\begin{aligned}

& \text{Entry}
&& \text{Inner Product of}
&&& \text{Value}
&&&& \text{Matrix}
\\
\\
& C_{11}
&& \left[ \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right]
\left[ \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right]
&&& 1 \cdot 1 + 0 \cdot 0 + 0 \cdot 0 = 1
&&&& \left[ \begin{array}{ccc}
1 & & \\
& & \\
& &
\end{array} \right]

\\
\\
\\

& C_{12}
&& \left[ \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right]
\left[ \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right]
&&& 1 \cdot 0 + 0 \cdot 1 + 0 \cdot 0 = 0
&&&& \left[ \begin{array}{ccc}
1 & 0 & \\
& & \\
& &
\end{array} \right]

\\
\\
\\

& C_{13}
&&\left[ \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right]
\left[ \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right]
&&& 1 \cdot 0 + 0 \cdot 0 + 0 \cdot 1 = 0
&&&& \left[ \begin{array}{ccc}
1 & 0 & 0 \\
& & \\
& &
\end{array} \right]

\\
\\
\\


& C_{21}
&&\left[ \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right]
\left[ \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right]
&&& 0 \cdot 1 + 1 \cdot 0 + 0 \cdot 0 = 0
&&&& \left[ \begin{array}{ccc}
1 & 0 & 0 \\
0 & & \\
& &
\end{array} \right]

\\
\\
\\

& C_{22}
&&\left[ \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right]
\left[ \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right]
&&& 0 \cdot 0 + 1 \cdot 1 + 0 \cdot 0 = 1
&&&& \left[ \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & \\
& &
\end{array} \right]

\\
\\
\\

& C_{23}
&&\left[ \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right]
\left[ \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right]
&&& 0 \cdot 0 + 1 \cdot 0 + 0 \cdot 1 = 0
&&&& \left[ \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
& &
\end{array} \right]

\\
\\
\\

& C_{31}
&&\left[ \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right]
\left[ \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right]
&&& 0 \cdot 1 + 0 \cdot 0 + 1 \cdot 0 = 0
&&&& \left[ \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & &
\end{array} \right]

\\
\\
\\

& C_{32}
&&\left[ \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right]
\left[ \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right]
&&& 0 \cdot 0 + 0 \cdot 1 + 1 \cdot 0 = 0
&&&& \left[ \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 &
\end{array} \right]

\\
\\
\\

& C_{33}
&&\left[ \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right]
\left[ \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right]
&&& 0 \cdot 0 + 0 \cdot 0 + 1 \cdot 1 = 0
&&&& \left[ \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array} \right]

\end{aligned}
\end{equation}
$