Question

A restaurant has a main location and a traveling food truck. The first matrix A shows the number of managers and associates employed. The second matrix B shows the average annual cost of salary and benefits​ (in thousands of​ dollars). Complete parts​ (a) through​ (c) below.

Managers Associates
Restaurant 5 25 = A
Food Truck 1 4

Salary Benefits
Managers 41 6 = B
Associates 20 2

a. Find the matrix product AB .
b. Explain what AB represents.
c. According to matrix AB , what is the total cost of salaries for all employees (managers and associates) at the restaurant? What is the total cost of benefits for all employees at the food truck?

Answer 1

A*B=
`\left[\begin{array}{cc}705&80\\121&14 \end{array}\right]`

Explanation:

Given A=
`\left[\begin{array}{cc}5&25\\1&4\end{array}\right] \left[\begin{array}{cc}41&6\\20&2\end{array}\right]` = B

Finding A*B means multiplying the first row with the first column and first row with the second column would give the first row elements. The second ro0w elements are obtained by multiplying the second row with the 1st column and second row with the second column.

so A*B=
`\left[\begin{array}{cc}5*41+ 25*20&5*6 + 25*2\\ 1*41+4*20 & 1*6+ 4*2\end{array}\right]`

Now multiply and add the separate elements of the matrix A*B=

`\left[\begin{array}{cc}205+500&30+50\\41+80&6+8\end{array}\right]`

A*B=
`\left[\begin{array}{cc}705&80\\121&14 \end{array}\right]`

b. The 1st element of the 1st row shows the salaries of the managers and 2nd element of the 1st row the salaries of associates at the restaurant . The second row 1 st element shows the benefits of the managers and 2nd element the benefits of the associates at the food truck.

c. The total cost of salaries for all employees (managers and associates) at the restaurant = 705 + 80 = 785

Total cost of benefits for all employees at the food truck= 121 + 14= 135