College Algebra, Chapter 7, 7.2, Section 7.2, Problem 34

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.) $DB + DC$


$
\begin{equation}
\begin{aligned}

DB + DC =& \left[ \begin{array}{cc}
7 & 3 \end{array} \right]

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

+
\left[ \begin{array}{cc}
7 & 3 \end{array} \right]

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

\\
\\
\\

=& \left[ \begin{array}{ccc}
7 \cdot 3 + 3 \cdot 1 & \displaystyle 7 \cdot \frac{1}{2} + 3 \cdot (-1) & 7 \cdot 5 + 3 \cdot 3
\end{array} \right]

\left[ \begin{array}{ccc}
7 \cdot 2 + 3 \cdot 0 & \displaystyle 7 \cdot \left( \frac{-5}{2} \right) + 3 \cdot 2 & 7 \cdot 0 + 3 \cdot (-3)
\end{array} \right]

\\
\\
\\

=&
\left[ \begin{array}{ccc}
24 & \displaystyle \frac{1}{2} & 44
\end{array} \right]
+

\left[ \begin{array}{ccc}
14 & \displaystyle - \frac{23}{2} & -9
\end{array} \right]


\end{aligned}
\end{equation}
$


b.) $BF + FE$


$
\begin{equation}
\begin{aligned}

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

\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]

\left[ \begin{array}{ccc}
1\\
2\\
0
\end{array} \right]

\\
\\
\\

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

+

\left[ \begin{array}{c}
1 \cdot 1 + 0 \cdot 2 + 0 \cdot 0 \\
0 \cdot 1 + 1 \cdot 2 + 0 \cdot 0 \\
0 \cdot 1 + 0 \cdot 2 + 1 \cdot 0
\end{array} \right]

\\
\\
\\

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

\end{aligned}
\end{equation}
$


$BF + FE$ is not undefined because we cannot add matrices of different dimensions.

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