Chapter 8.2, Problem 71E

(A)

To determine

To find:

The product of two matrices $AB$

Answer to Problem 71E

The product of two matrices $AB={\left[\begin{array}{l}\begin{array}{cc}\$571.8& \$206.6\\ \$798.9& \$288.8\end{array}\\ \begin{array}{cc}\$936& \$337.8\end{array}\end{array}\right]}_{3\times 2}$

Explanation of Solution

Given:

The matrix

  $\begin{array}{l}A={\left[\begin{array}{ccc}40& 64& 52\\ 60& 82& 76\\ 76& 96& 84\end{array}\right]}_{3\times 3},B={\left[\begin{array}{l}\begin{array}{cc}\$3.45& \$1.20\\ \$3.65& \$1.30\end{array}\\ \begin{array}{cc}\$3.85& \$1.45\end{array}\end{array}\right]}_{3\times 2}\\ \end{array}$

Concept used:

Definition of matrix multiplication :-If A $=\left[a_{ij}\right]$ is an $m\times n$ matrix and B $=\left[b_{ij}\right]$ is an $n\times p$ matrix , then the product $ST$ is an $m\times p$ matrix

  $\begin{array}{l}AB=\left[a_{ij}\times b_{ij}\right]\\ AB=\left[c_{ij}\right]\end{array}$

Where $c_{ij}$ = $=a_{i1}{b}_{1j}+a_{i2}{b}_{2j}+a_{i3}{b}_{3j}+\mathrm{..........}+a_{in}{b}_{nj}$

Calculation:

The product $AB$ is an $m\times p$ matrix

  $\begin{array}{l}AB=\left[a_{ij}\times b_{ij}\right]\\ AB=\left[c_{ij}\right]\end{array}$

Where $c_{ij}$ = $=a_{i1}{b}_{1j}+a_{i2}{b}_{2j}+a_{i3}{b}_{3j}+\mathrm{..........}+a_{in}{b}_{nj}$

The matrix

  $\begin{array}{l}A={\left[\begin{array}{ccc}40& 64& 52\\ 60& 82& 76\\ 76& 96& 84\end{array}\right]}_{3\times 3},B={\left[\begin{array}{l}\begin{array}{cc}\$3.45& \$1.20\\ \$3.65& \$1.30\end{array}\\ \begin{array}{cc}\$3.85& \$1.45\end{array}\end{array}\right]}_{3\times 2}\\ \end{array}$

The first column corresponds to plant S and the second column corresponds to plant

  $\begin{array}{l}AB={\left[\begin{array}{ccc}40& 64& 52\\ 60& 82& 76\\ 76& 96& 84\end{array}\right]}_{3\times 3}{\left[\begin{array}{l}\begin{array}{cc}\$3.45& \$1.20\\ \$3.65& \$1.30\end{array}\\ \begin{array}{cc}\$3.85& \$1.45\end{array}\end{array}\right]}_{3\times 2}\\ AB={\left[\begin{array}{l}\begin{array}{cc}\$571.8& \$206.6\\ \$798.9& \$288.8\end{array}\\ \begin{array}{cc}\$936& \$337.8\end{array}\end{array}\right]}_{3\times 2}\end{array}$

(B)

To determine

To find:

The dairy mart’s total profit from milk sales for the weekend

Answer to Problem 71E

The dairy mart’s total profit from milk sales for the weekend is $\$833.20$

Explanation of Solution

Given:

The matrix

  $\begin{array}{l}A={\left[\begin{array}{ccc}40& 64& 52\\ 60& 82& 76\\ 76& 96& 84\end{array}\right]}_{3\times 3},B={\left[\begin{array}{l}\begin{array}{cc}\$3.45& \$1.20\\ \$3.65& \$1.30\end{array}\\ \begin{array}{cc}\$3.85& \$1.45\end{array}\end{array}\right]}_{3\times 2}\\ \end{array}$

Concept used:

Definition of matrix multiplication :-If A $=\left[a_{ij}\right]$ is an $m\times n$ matrix and B $=\left[b_{ij}\right]$ is an $n\times p$ matrix , then the product $ST$ is an $m\times p$ matrix

  $\begin{array}{l}AB=\left[a_{ij}\times b_{ij}\right]\\ AB=\left[c_{ij}\right]\end{array}$

Where $c_{ij}$ = $=a_{i1}{b}_{1j}+a_{i2}{b}_{2j}+a_{i3}{b}_{3j}+\mathrm{..........}+a_{in}{b}_{nj}$

Calculation:

The product $AB$ is an $m\times p$ matrix

  $\begin{array}{l}AB=\left[a_{ij}\times b_{ij}\right]\\ AB=\left[c_{ij}\right]\end{array}$

Where $c_{ij}$ = $=a_{i1}{b}_{1j}+a_{i2}{b}_{2j}+a_{i3}{b}_{3j}+\mathrm{..........}+a_{in}{b}_{nj}$

The matrix

  $\begin{array}{l}A={\left[\begin{array}{ccc}40& 64& 52\\ 60& 82& 76\\ 76& 96& 84\end{array}\right]}_{3\times 3},B={\left[\begin{array}{l}\begin{array}{cc}\$3.45& \$1.20\\ \$3.65& \$1.30\end{array}\\ \begin{array}{cc}\$3.85& \$1.45\end{array}\end{array}\right]}_{3\times 2}\\ \end{array}$

The first column corresponds to plant S and the second column corresponds to profit from milk sales

  $\begin{array}{l}AB={\left[\begin{array}{ccc}40& 64& 52\\ 60& 82& 76\\ 76& 96& 84\end{array}\right]}_{3\times 3}{\left[\begin{array}{l}\begin{array}{cc}\$3.45& \$1.20\\ \$3.65& \$1.30\end{array}\\ \begin{array}{cc}\$3.85& \$1.45\end{array}\end{array}\right]}_{3\times 2}\\ AB={\left[\begin{array}{l}\begin{array}{cc}\$571.8& \$206.6\\ \$798.9& \$288.8\end{array}\\ \begin{array}{cc}\$936& \$337.8\end{array}\end{array}\right]}_{3\times 2}\end{array}$

The dairy mart’s total profit from milk sales for the weekend is $\$833.20$