Lesson 2 … and Row by Column Solidify Understanding

Jump Start

Multiply the two matrices together to find the product matrix:

$[\begin{matrix} 3 & 2 \\ 2 & 1 \end{matrix}] \cdot [\begin{matrix} 2 & 3 & 2 \\ 4 & 2 & 1 \end{matrix}]$

Learning Focus

Use matrix multiplication in context.

How do I use matrix multiplication to model contexts that depend on summing up multiple partial products?

Open Up the Math: Launch, Explore, Discuss

In the previous lesson, you chose a context and wrote a story for the information given in the following augmented matrix:

$[\begin{matrix} \begin{array}{r} 1 & 2 & 2 \\ 2 & 1 & 3 \\ 1 & 2 & 1 \end{array} & | \begin{array}{r} 16 \\ 17 \\ 13 \end{array} \end{matrix}]$

Your story could also have been represented by the following system of equations:

$\begin{matrix} x + 2 y + 2 z = 16 \\ 2 x + y + 3 z = 17 \\ x + 2 y + z = 13 \end{matrix}$ Using row reduction on the matrix, you found that the solution to this system of equations is:

$x = 2$ , $y = 4$ , $z = 3$ . In a later task in this unit, you will learn another method for solving linear systems using matrices. This new method will use matrix multiplication, so let’s review that operation.

1.

Use the following to review how matrix multiplication works.

a.

Explain how this matrix equation given represents the system of equations previously given.

$[\begin{matrix} 1 & 2 & 2 \\ 2 & 1 & 3 \\ 1 & 2 & 1 \end{matrix}] \cdot [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 16 \\ 17 \\ 13 \end{matrix}]$

b.

Now verify that $x = 2$ , $y = 4$ , $z = 3$ is a solution to the system of equations given above using matrix multiplication by replacing $x$ , $y$ , and $z$ with the proposed solution and carrying out the matrix multiplication.

$[\begin{matrix} 1 & 2 & 2 \\ 2 & 1 & 3 \\ 1 & 2 & 1 \end{matrix}] \cdot [\begin{matrix} 2 \\ 4 \\ 3 \end{matrix}] = [\begin{matrix} 16 \\ 17 \\ 13 \end{matrix}]$

2.

In a context, each entry in a matrix represents two pieces of information, depending on the row and column in which it is located. Organize the following information into a matrix, and label the rows and columns.

Week 1: Clarita painted $6$ curbside logos and Carlos painted $4$ driveway mascots.
Week 2: Clarita painted $8$ curbside logos and Carlos painted $3$ driveway mascots.
Week 3: Clarita painted $5$ curbside logos and Carlos painted $6$ driveway mascots.

3.

Carlos and Clarita charge $$ 8$ for a curbside logo and $$ 20$ for a driveway mascot. Using this additional information, show how you can use matrix multiplication to determine how much revenue Carlos and Clarita collected during weeks 1, 2, and 3.

Ready for More?

A photographer sells framed photos in $11 \times 14 in$ , $8 \times 10 in$ , and $5 \times 7 in$ sizes. His expenses include the cost for paper, ink, and framing material, as well as paying his staff for assembly time. The photographer sells the most around Mother’s Day and Thanksgiving, and would like to know his assembly time and costs for these two holidays.

During the weeks prior to Mother’s Day the photographer uses $10$ reams of paper, $12$ sets of inkjet cartridges, and $200$ feet of framing material. Around Thanksgiving, the photographer uses $24$ reams of paper, $20$ sets of inkjet cartridges, and $360$ feet of framing material. A ream of photo paper costs $$ 25$ , a set of inkjet cartridges costs $$ 90$ , and framing material costs $$ 2$ per foot.

Assembling the framed photos includes cutting the paper at $4$ hours per ream, replacing the ink at $\frac{1}{2}$ hour per set of inkjet cartridges, and building the frames, which averages $\frac{3}{4}$ hour per foot of framing material used.

Use this information to find the total cost and assembly time for each holiday using matrix multiplication.

Takeaways

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

For example, these two matrices can be multiplied together:

$Matrix A = [\begin{matrix} 8 & 2 & 1 \\ 10 & 4 & 3 \end{matrix}], Matrix B = [\begin{matrix} 2 & 2 \\ 3 & 4 \\ 2 & 1 \end{matrix}]$

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

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

Lesson Summary

In this lesson, we reviewed the row-by-column procedure for multiplying matrices, and used matrix multiplication to model a context.

Retrieval

1.

Add the matrices.

$[\begin{array}{rrr} 3 & - 7 & 9 \\ - 5 & 4 & - 8 \end{array}] + [\begin{array}{rrr} 2 & 10 & - 2 \\ 5 & - 11 & 3 \end{array}] =$

2.

Create two matrices that cannot be added.

3.

What is the meaning of $P (A | B)$ ?

4.

What is the meaning of $P (A \cup B)$ ?