Lesson 8 Eating Up the Lunchroom Budget Develop Understanding

Learning Focus

Multiply matrices to model a context.

How does the row and column structure of the matrices facilitate the work of matrix multiplication?

When multiplying matrices, how do the values in the factor matrices relate and connect to one another in order to produce a meaningful result?

What processes seem to be repeated and systematic in the work of calculating data representing the product of two matrices?

Technology guidance for today’s lesson:

Multiply Matrices: Casio ClassPad Casio fx-9750GIII

Open Up the Math: Launch, Explore, Discuss

Here is a matrix representing the number of food items Elvira plans to order this year for the soccer team and drama club celebrations. (Note: Labels have been added to keep track of the meaning of the rows and columns.)

$\begin{array}{rcc} \begin{array}{ccc} chips & cookies & drinks \end{array} \\ \begin{array}{rcc} Soccer \\ Drama \end{array} & [\begin{array}{cccc} 8 & 7 & 4 \\ 10 & 4 & 4 \end{array}] \end{array}$

Here is a matrix representing the cost of purchasing each food item at two different stores, Main Street Market and Grandpa’s Grocery.

$\begin{array}{rc} \begin{array}{cc} Main Street & Grandpa’s \\ Market & Grocery \end{array} \\ \begin{array}{rcc} Cost per package of chips \\ Cost per dozen cookies \\ Cost per gallon of drink \end{array} & [\begin{array}{cccc} $ 2.50 & $ 2.00 \\ $ 3.00 & $ 4.00 \\ $ 2.00 & $ 1.50 \end{array}] \end{array}$

Help Elvira determine how much each event would cost if all the food for the event was purchased at Main Street Market or Grandpa’s Grocery. These total amounts could be recorded in a matrix that looks like this:

$\begin{array}{rc} \begin{array}{cc} Main Street & Grandpa’s \\ Market & Grocery \end{array} \\ \begin{array}{rcc} Soccer \\ Drama \end{array} & [\begin{array}{cccc} a & b \\ c & d \end{array}] \end{array}$

1.

Calculate the values of $a$ , $b$ , $c$ , and $d$ in the matrix provided previously.

2.

Explain, in detail, how you would use the numbers in the first two matrices above to obtain the values for the third matrix.

3.

In addition to the soccer team and drama club, Elvira plans to host events for the chess club, the cheerleaders, and the football team. She gives you the following matrix to represent food items that need to be ordered for each event. Can you use matrix multiplication with the cost matrix given above to determine the total cost of each event if items are purchased at each store? If yes, show how. If no, explain why not.

$\begin{array}{cc} \begin{matrix} \end{matrix} & \begin{matrix} chips & cookies & drinks \end{matrix} \\ \begin{matrix} Soccer \\ Drama \\ Chess \\ Cheerleaders \\ Football \end{matrix} & [\begin{matrix} 8 & 7 & 4 \\ 10 & 4 & 4 \\ 3 & 4 & 2 \\ 2 & 3 & 2 \\ 14 & 12 & 8 \end{matrix}] \end{array}$

4.

In addition to chips, cookies, and drinks, Elvira plans to add rolls and cold cuts to the events’ menu. She gives you the following matrix to represent all of the food items that need to be ordered for each event. Can you use matrix multiplication with the cost matrix given above to determine the total cost of each event if items are purchased at each store? If yes, show how. If no, explain why not.

$\begin{array}{cc} \begin{matrix} \end{matrix} & \begin{matrix} chips & cookies & drinks & rolls & cold cuts \end{matrix} \\ \begin{matrix} Soccer \\ Drama \\ Chess \\ Cheerleaders \\ Football \end{matrix} & [\begin{matrix} 8 & 7 & 4 & 6 & 4 \\ 10 & 4 & 4 & 8 & 5 \\ 3 & 4 & 2 & 2 & 2 \\ 2 & 3 & 2 & 2 & 2 \\ 14 & 12 & 8 & 12 & 10 \end{matrix}] \end{array}$

Ready for More?

Modify the scenario in problem 4 so that you can solve Elvira’s new problem. What information are you missing? How would you organize that information into a matrix? Use the information you provide to calculate the purchase price for each event, depending on the store where you make the purchases. What recommendations would you make for Elvira?

Takeaways

Matrices can be multiplied together if their dimensions are compatible. This means

For example, these two matrices can be multiplied together:

Matrix $A$ =

Matrix $B$ =

When used to model a context, matrices can be multiplied together if their units are compatible. This means

Revise and Refine:

Using the matrices given above, describe and illustrate how to find the matrix product $A \times B$ :

Vocabulary

matrix multiplication
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned how to perform the operation of matrix multiplication. The row-and-column structure of the two factor matrices facilitates this work since every element in the product matrix comes from adding together several partial products.

Retrieval

The pairs of equations are equivalent. Determine what was done to the first equation to get the second equation.

1.

$\begin{matrix} 2 x + 2 y = 10 \\ - 4 x - 4 y = - 20 \end{matrix}$

2.

$\begin{matrix} 3 x + 4 y = 18 \\ 4 y = 18 - 3 x \end{matrix}$

Determine whether the functions represented are linear, exponential, or neither. State what type of function and explain your answer.

3.

$x$	$f (x)$
$- 2$	$128$
$- 1$	$64$
$0$	$32$
$1$	$16$
$2$	$8$
$3$	$4$

4.

$g (x) = - 7 (x - 2) + 5$