College Algebra, Chapter 7, 7.2, Section 7.2, Problem 48

A farmer's three children, Amy, Beth and Chad, run three roadside produce stands during the summer months. One weekend they all sell watermelons, yellow squash and tomatoes. The matrices $A$ and $B$ tabulate the number of pounds of each product sold by each sibling on Saturday and Sunday














The matrix $C$ gives the price per pound (in dollars) for each type of produce that they sell.







Perform each of the following matrix operations, and interpret the entries in each result.

a.) $AC \qquad$ b.) $BC \qquad$ c.) $A+B \qquad$ d.) $(A+B)C$

a.) $\displaystyle AC = \left[ \text{Matrix A} \right] \left[ \text{Matrix B} \right]
= \left[ \begin{array}{c}
120(0.10) + 50(0.50) + 60(1.00) \\
40(0.10) + 25(0.50) + 30(1.00) \\
60(0.10) + 30 (0.50) + 20(1.00)
\end{array} \right]
=
\left[ \begin{array}{c}
97 \\
46.5 \\
41
\end{array} \right]
$

When we take the inner product of row in $A$ with a column in $B$, we are multiplying the total number of weight to its corresponding unit price per weight. Thus, $AC$ shows the profit of the siblings on Saturday.

The values $97,46.5$ and $41$ are the profit of Amy, Beth and Chad respectively.

b.) $\displaystyle BC = \left[ \text{Matrix B} \right] \left[ \text{Matrix C} \right] = \left[ \begin{array}{c}
100(0.10) + 60(0.50) + 30(1.00) \\
35(0.10) + 20(0.50) + 20(1.00) \\
60 (0.10) + 25(0.50) + 30(1.00)
\end{array} \right] =
\left[ \begin{array}{c}
70 \\
33.5 \\
48.5
\end{array} \right]
$

Same as from part (b). But its the sibling's profit on Sunday.

c.) $\displaystyle A + B = \left[ \text{Matrix A} \right] + \left[ \text{Matrix B} \right] = \left[ \begin{array}{ccc}
120 +100 & 50+60 & 60+30 \\
40+35 & 25+20 & 30+20 \\
60+60 & 30+25 & 20+30
\end{array} \right] = \left[ \begin{array}{ccc}
220 & 110 & 90 \\
75 & 45 & 50 \\
120 & 55 & 50
\end{array} \right]$

This shows the total number of weight sold by each child on Saturday and Sunday.

d.) $\displaystyle (A+ B)C = \left[ \begin{array}{ccc}
220 & 110 & 90 \\
75 & 45 & 50 \\
120 & 55 & 50
\end{array} \right] \left[ \begin{array}{c}
0.10 \\
0.50 \\
1.00
\end{array} \right] = \left[ \begin{array}{c}
220(0.10) + 110(0.50) + 90(1.00) \\
75(0.10) + 45(0.50) + 50(1.00) \\
120(0.10) + 55 (0.50) + 50 (1.00)
\end{array} \right]=
\left[ \begin{array}{c}
167 \\
80 \\
89.5
\end{array} \right]
$

This shows the total profit from each child on Saturday and Sunday. The values $167,80$ and $89.5$ are the profit of Amy, Beth and Chad respectively.