Chapter 8.2, Problem 69E

To determine

To find:

The product of two matrices $AB$

Answer to Problem 69E

The product of two matrices $BA={\left[\begin{array}{cc}\$17,699,050& \$17,299,050\end{array}\right]}_{1\times 2}$

Explanation of Solution

Given:

The matrix

  $\begin{array}{l}A={\left[\begin{array}{l}\begin{array}{cc}5,000& 4,000\\ 6,000& 10,000\end{array}\\ \begin{array}{cc}8,000& 5,000\end{array}\end{array}\right]}_{3\times 2}\\ B={\left[\begin{array}{ccc}\$699.95& \$899.95& \$1099.95\end{array}\right]}_{1\times 3}\end{array}$

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

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

  $\begin{array}{l}A={\left[\begin{array}{l}\begin{array}{cc}5,000& 4,000\\ 6,000& 10,000\end{array}\\ \begin{array}{cc}8,000& 5,000\end{array}\end{array}\right]}_{3\times 2}\\ B={\left[\begin{array}{ccc}\$699.95& \$899.95& \$1099.95\end{array}\right]}_{1\times 3}\end{array}$

The product of two matrices $AB$

  $\begin{array}{l}BA={\left[\begin{array}{ccc}\$699.95& \$899.95& \$1099.95\end{array}\right]}_{1\times 3}{\left[\begin{array}{l}\begin{array}{cc}5,000& 4,000\\ 6,000& 10,000\end{array}\\ \begin{array}{cc}8,000& 5,000\end{array}\end{array}\right]}_{3\times 2}\\ BA={\left[\begin{array}{cc}\$17,699,050& \$17,299,050\end{array}\right]}_{1\times 2}\end{array}$

The matrix $BA$ gives the total price of all units shipped to each warehouse