Chapter 10.4, Problem 48E

Produce Sales 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.

$\begin{array}{rr}\hfill & \hfill Priceperpound\\ \hfill \begin{array}{r}\hfill Melons\\ \hfill Squash\\ \hfill Tomatoes\end{array}& \hfill \left[\begin{array}{r}\hfill 0.10\\ \hfill 0.50\\ \hfill 1.00\end{array}\right]=C\end{array}$

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

1. (a) AC
2. (b) BC
3. (c) A + B
4. (d) (A + B)C

To determine

To evaluate: The product of matrix A and C and interpret the entries of matrix AC.

Answer to Problem 48E

The product of matrix A with matrix C is $\left[\begin{array}{c}97\\ 62.5\\ 41\end{array}\right]$ and total price of products sold by Amy on Saturday is $97, by Beth is $62.5 and by Chad is $41.

Explanation of Solution

Given:

The given matrices are, $A=\left[\begin{array}{ccc}120& 50& 60\\ 40& 25& 30\\ 60& 30& 20\end{array}\right]$ and $C=\left[\begin{array}{c}0.10\\ 0.50\\ 1.00\end{array}\right]$.

Calculation:

Multiply matrix A with matrix C,

$\begin{array}{c}AC\text{=}\left[\begin{array}{ccc}120& 50& 60\\ 40& 25& 30\\ 60& 30& 20\end{array}\right]\left[\begin{array}{c}0.10\\ 0.50\\ 1.00\end{array}\right]\\ =\left[\begin{array}{c}12+25+60\\ 20+12.5+30\\ 6+15+20\end{array}\right]\\ =\left[\begin{array}{c}97\\ 62.5\\ 41\end{array}\right]\end{array}$

Hence, the product of matrix A with matrix C is $\left[\begin{array}{c}97\\ 62.5\\ 41\end{array}\right]$.

Matrix A tabulates the number of pound of each product sold by each sibling on Saturday,

Rows of A tabulate the number of pounds of product sold by each siblings and column of A shows the type of products. That is Melon, Yellow squash, Tomatoes.

And matrix, C gives the price per pound (in dollars) for each type of product that they sale.

The rows of this matrix AC total price of products sold by each siblings on Saturday and columns of AC give the total price of each product.

Hence, total price of products sold by Amy on Saturday is $97, by Beth is $62.5 and by Chad is $41.

To determine

To evaluate: The product of matrix B and C and interpret the entries of matrix BC.

Answer to Problem 48E

The product of matrix B with matrix C is $\left[\begin{array}{c}70\\ 33.5\\ 48.5\end{array}\right]$ and total price of products sold by Amy on Saturday is $70, by Beth is $33.5 and by Chad is $48.5.

Explanation of Solution

Given:

The given matrices are,

$\begin{array}{l}B=\left[\begin{array}{ccc}100& 60& 30\\ 35& 20& 20\\ 60& 25& 30\end{array}\right]\\ C=\left[\begin{array}{c}0.10\\ 0.50\\ 1.00\end{array}\right]\end{array}$

Calculation:

Multiply matrix B with matrix C

$\begin{array}{c}BC\text{=}\left[\begin{array}{ccc}100& 60& 30\\ 35& 20& 20\\ 60& 25& 30\end{array}\right]\left[\begin{array}{c}0.10\\ 0.50\\ 1.00\end{array}\right]\\ =\left[\begin{array}{c}10+30+30\\ 3.5+10+20\\ 6+12.5+30\end{array}\right]\\ =\left[\begin{array}{c}70\\ 33.5\\ 48.5\end{array}\right]\end{array}$

Hence, the product of matrix B with matrix C is $\left[\begin{array}{c}70\\ 33.5\\ 48.5\end{array}\right]$

Matrix B tabulates the number of pound of each product sold by each sibling on Sunday,

Rows of B tabulate the number of pounds of product sold by each sibling. And column of B shows the type of products. That is Melon, Yellow squash, Tomatoes.

And matrix, C gives the price per pound (in dollars) for each type of product that they sale.

The rows of this matrix AC total price of products sold by each siblings on Saturday and columns of AC give the total price of each product.

Hence, total price of products sold by Amy on Saturday is $70, by Beth is $33.5 and by Chad is $48.5.

To determine

To evaluate: The addition of matrix A and B and interpret entries of matrix A+C.

Answer to Problem 48E

The addition of matrix A with matrix B is $\left[\begin{array}{ccc}220& 110& 90\\ 75& 45& 50\\ 120& 55& 50\end{array}\right]$ and in two days Amy sold 220 pounds of Melon, 110 pounds of Yellow squash, and 90 pounds of Tomatoes. Beth sold 75 pounds of Melon 45pounds of Yellow squash, and 50 pounds of Tomatoes. Chad sold 120 pound of Melon, 55 pound of Yellow squash, and 50 pounds of Tomatoes.

Explanation of Solution

Given:

The given matrices are,

$\begin{array}{l}A=\left[\begin{array}{ccc}120& 50& 60\\ 40& 25& 30\\ 60& 30& 20\end{array}\right]\\ B=\left[\begin{array}{ccc}100& 60& 30\\ 35& 20& 20\\ 60& 25& 30\end{array}\right]\\ C=\left[\begin{array}{c}0.10\\ 0.50\\ 1.00\end{array}\right]\end{array}$

Calculation:

Add matrix A with matrix B,

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

Hence, the addition of matrix A with matrix B is $\left[\begin{array}{ccc}220& 110& 90\\ 75& 45& 50\\ 120& 55& 50\end{array}\right]$

Matrix A and B tabulates the number of pound of each product sold by each sibling on Saturday and Sunday respectively,

Rows of A and B tabulate the number of pounds of product sold by each sibling on Saturday and Sunday respectively.

And column of A and B shows the type of products. That is Melon, Yellow squash, Tomatoes.

The rows of this matrix A+B tabulates the total number of pound of each product sold by each sibling on Saturday and Sunday.

And columns of A+B show the type of products. That is Melon, Yellow squash, Tomatoes.

Hence, in two days Amy sold 220 pounds of Melon, 110 pounds of Yellow squash, and 90 pounds of Tomatoes. Beth sold 75 pounds of Melon 45pounds of Yellow squash, and 50 pounds of Tomatoes. Chad sold 120 pound of Melon, 55 pound of Yellow squash, and 50 pounds of Tomatoes.

To determine

To evaluate: The matrix operation $\left(A+B\right)C$ and interpret entries of matrix A+C.

Answer to Problem 48E

The value matrix $\left(A+B\right)C$ is $\left[\begin{array}{c}167\\ 80\\ 89.5\end{array}\right]$ and total price of products sold by Amy in two days is $167, by Beth is $80 and by Chad is $89.5.

Explanation of Solution

Given:

The given matrices are,

$\begin{array}{l}A=\left[\begin{array}{ccc}120& 50& 60\\ 40& 25& 30\\ 60& 30& 20\end{array}\right]\\ B=\left[\begin{array}{ccc}100& 60& 30\\ 35& 20& 20\\ 60& 25& 30\end{array}\right]\\ C=\left[\begin{array}{c}0.10\\ 0.50\\ 1.00\end{array}\right]\end{array}$

Calculation:

Add matrix A with matrix B,

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

Now multiply $\left(A+B\right)$ with C.

$\begin{array}{c}\left(A+B\right)C=\left[\begin{array}{ccc}220& 110& 90\\ 75& 45& 50\\ 120& 55& 50\end{array}\right]\left[\begin{array}{c}0.10\\ 0.50\\ 1.00\end{array}\right]\\ =\left[\begin{array}{c}22+55+90\\ 7.5+22.5+50\\ 12+27.5+50\end{array}\right]\\ =\left[\begin{array}{c}167\\ 80\\ 89.5\end{array}\right]\end{array}$

Hence, the value matrix $\left(A+B\right)C$ is $\left[\begin{array}{c}167\\ 80\\ 89.5\end{array}\right]$.

Matrix A and B tabulates the number of pound of each product sold by each sibling on Saturday and Sunday respectively,

Rows of A and B tabulate the number of pounds of product sold by each sibling on Saturday and Sunday respectively.

And column of A and B shows the type of products. That is Melon, Yellow squash, Tomatoes.

The rows of this matrix (A+B)C tabulates then total price of products sold by each siblings on Saturday and Sunday and columns of (A+B)C tabulates the total price of each product.

Hence, total price of products sold by Amy in two days is $167, by Beth is $80 and by Chad is $89.5.