Lesson 10 The Arithmetic of Matrices Practice Understanding

Learning Focus

Model contexts with matrices.

How are matrix operations applied to real-world problems?

In which contexts are the different operations appropriately used?

Open Up the Math: Launch, Explore, Discuss

Many clubs do not have the funds to pay for parties and events in the cafeteria. Therefore, Elvira lets club members, and their parents, volunteer for service hours and gives each club credit for the amount of volunteer hours they provide. There are three types of chores the volunteers can do: setting up tables, mopping the floors, and washing dishes. They can volunteer for weekday hours or for weekend hours, which earn more credits.

Elvira has recorded volunteer hours for the month of September for the drama club and the chess club in the following matrices:

Drama Club

$\begin{array}{cc} \begin{matrix} \end{matrix} & \begin{matrix} weekdays & weekends \end{matrix} \\ \begin{matrix} tables \\ floors \\ dishes \end{matrix} & [\begin{matrix} 12 & 4 \\ 8 & 3 \\ 6 & 3 \end{matrix}] \end{array}$

Chess Club

1.

Write and solve a matrix equation to find the total weekday and weekend volunteer hours for each type of chore.

The drama club has committed to provide the same number of volunteer hours each month for all $9$ months of the school year.

2.

Write and solve a matrix equation that gives the total weekday and weekend volunteer hours for each type of chore that will be provided during the school year by the drama club.

Because it is harder to get volunteers for weekends than for weekdays, Elvira gives more credit for weekend hours than for weekday hours. She credits $$ 4$ per hour for volunteer hours performed on weekdays, and $$ 6$ per hour on weekends.

3.

Write and solve a matrix equation to find the total credit earned by the chess club during September for each type of chore.

4.

Find the total amount of money Elvira credited for volunteer hours by the two clubs combined during the month of September.

Elvira is getting good at manipulating matrices but realizes that sometimes she only needs one element in the sum or product matrix (for example, the cost of buying ingredients at Grandpa’s Grocery on a specific day) so she would like to be able to calculate a single result without completing the rest of the matrix operation. For the following matrix operations, calculate the indicated missing elements in the sum or product, without calculating the rest of the individual elements in the sum or product matrix.

5.

$[\begin{matrix} 5 & - 2 & 3 & 6 \\ 7 & 1 & - 4 & 2 \end{matrix}] + [\begin{matrix} 1 & 3 & 5 & - 7 \\ 4 & - 3 & 2 & 5 \end{matrix}] = [\begin{matrix} - & - & ◻ & - \\ - & ◻ & - & - \end{matrix}]$

6.

$[\begin{matrix} - 2 & 3 \\ 4 & 1 \\ 2 & 5 \\ 1 & 3 \end{matrix}] \times [\begin{matrix} 2 & - 3 & 4 \\ - 1 & 5 & - 2 \end{matrix}] = [\begin{matrix} - & ◻ & - \\ ◻ & - & - \\ - & - & ◻ \\ - & - & - \end{matrix}]$

7.

$3 \times [\begin{matrix} 2 & 4 \\ - 1 & 5 \end{matrix}] - 4 \times [\begin{matrix} 2 & - 3 \\ - 5 & 4 \end{matrix}] = [\begin{matrix} - & ◻ \\ ◻ & - \end{matrix}]$

8.

$[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & ◻ & 1 \end{matrix}] \times [\begin{matrix} 2 & 3 & ◻ \\ 1 & 2 & 4 \\ ◻ & 3 & 1 \end{matrix}] = [\begin{matrix} 2 & ◻ & 5 \\ 1 & 2 & ◻ \\ 4 & 3 & ◻ \end{matrix}]$

Given $A = [\begin{matrix} 2 & 1 \\ 3 & 4 \end{matrix}]$ and $B = [\begin{matrix} 2 & 4 \\ 1 & 3 \end{matrix}]$

9.

Does $A + B = B + A$ ? Will this be true for all matrices?

10.

Does $A \times B = B \times A$ ? Will this be true for all matrices?

Ready for More?

In Lesson 2, you reviewed properties of operations for addition and multiplication of numbers. Now that we have defined the operations of addition and multiplication for matrices, we can determine if these properties of operations still hold. Using examples or reasoning, decide which of the properties of addition and multiplication still work with matrices.

That is,

a.

Is matrix addition commutative? What about matrix multiplication?

b.

Is matrix addition associative? What about matrix multiplication?

c.

Is there an additive identity for matrices? What about a multiplicative identity?

d.

Does every matrix, except the identity matrix, have an additive inverse? What about a multiplicative inverse?

e.

Does the distributive property work for matrices?

Takeaways

Things I need to think about when modeling a situation with matrices:

Lesson Summary

In this lesson, we focused on writing and solving matrix equations to model different situations, including situations that involved matrix addition, matrix multiplication, the distributive property, and the scaling up of data. We found that the properties of operations can impact the way we write matrix equations, particularly when matrix multiplication is involved.

Retrieval

Graph the relationships that are given.

1.

$y = \frac{1}{2} x - 8$

2.

$x$	$y$
$- 2$	$3$
$0$	$7$
$1$	$9$
$3$	$13$

Evaluate each of the expressions using the given values of $x = 3$ , $y = - 2$ , and $z = 5$ .

3.

$3 x + 5 y$

4.

$- 2 (x + 5) - y + 2 z$

5.

$\frac{{(x + 2)}^{3}}{x z^{2}}$