Lesson 12 Solving Systems with Matrices Practice Understanding

Learning Focus

Apply a standard procedure for row-reduction of matrices.

How can I organize my work to make the row-reduction of a matrix more efficient?

Why is getting 1’s on the diagonal helpful?

Why do I want to get 0’s in the non-diagonal positions of one column before I move on to working with getting a 1 in the next column?

While not one of the “official” row-reduction steps, why might switching rows of the matrix be helpful?

Technology guidance for today’s lesson:

Equation Solving Using a Run Matrix: Casio fx-9750GIII

Open Up the Math: Launch, Explore, Discuss

In the task To Market with Matrices, you developed a strategy for solving systems of linear equations using matrices. An efficient and consistent way to carry out this strategy can be summarized as follows:

To row reduce a matrix:

Perform elementary row operations to yield a $1$ in the first row, first column.
Create $0$ ’s in all of the other rows of the first column by adding the first row times a constant to each other row.
Perform elementary row operations to yield a $1$ in the second row, second column.
Create $0$ ’s in all of the other rows of the second column by adding the second row times a constant to each other row.
Perform elementary row operations to yield a $1$ in the third row, third column.
Create $0$ ’s in all of the other rows of the third column by adding the third row times a constant to each other row.
Continue this process until the first $m \times m$ entries form a square matrix with $1$ ’s in the diagonal and $0$ ’s everywhere else.

Practice this strategy by creating a sequence of matrices for each of the following problems that begins with the given matrix and ends with the left portion of the matrix (the first $m \times m$ entries) in row-reduced form. Write a description of what you did to get from one matrix to another in each step of your sequence of matrices.

1.

$[\begin{matrix} 2 & 4 & 0 \\ 3 & 5 & - 2 \end{matrix}]$

2.

$[\begin{matrix} 4 & - 2 & 2 \\ 1 & 3 & 11 \end{matrix}]$

3.

$[\begin{matrix} 1 & 2 & 2 & 6 \\ 2 & 2 & 0 & 2 \\ 4 & 4 & 2 & 10 \end{matrix}]$

4.

Each of the previous matrices represents a system of equations. For each problem, write the system of equations represented by the original matrix. Determine the solution for each system using the reduced row echelon matrix you obtained, and then check the solutions in the original system.

5.

Solve the following problem by using a matrix to represent the system of equations described in the scenario, and then changing the matrix to reduced row echelon form to obtain the solution.

Three of Carlos’s and Clarita’s friends are purchasing school supplies at the bookstore. Stan buys $1$ notebook, $3$ packages of pencils, and $2$ markers for $$ 7.50$ . Jan buys $2$ notebooks, $6$ packages of pencils, and $5$ markers for $$ 15.50$ . Fran buys $1$ notebook, $2$ packages of pencils, and $2$ markers for $$ 6.25$ . How much do each of these $3$ items cost?

6.

Create a linear system that is either dependent (both equations in the system represent the same line) or inconsistent (the equations in the system represent non-intersecting lines). What happens when you try to row reduce the $2 \times 3$ matrix that represents this linear system of equations?

Ready for More?

A system of three equations with three unknowns was solved using row-reduction on a matrix. The row-reduction steps were performed in the following order, until the matrix shown on the right was obtained.

$\begin{matrix} - 2 R_{1} + R_{2} \to R_{2} \\ - 4 R_{1} + R_{3} \to R_{3} \\ \begin{matrix} - \frac{1}{2} R_{2} \to R_{2} \\ 4 R_{2} + R_{3} \to R_{3} \\ \begin{matrix} \frac{1}{2} R_{3} \to R_{3} \\ - 2 R_{3} + R_{2} \to R_{2} \\ 2 R_{3} + R_{1} \to R_{1} \end{matrix} \end{matrix} \end{matrix} ⟹ [\begin{matrix} 1 & 0 & 0 & - 1 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & 2 \end{matrix}]$

The matrix on the right shows the solutions to a system of equations. Find the original system of equations and associated matrix whose solutions are shown in this final matrix.

Takeaways

The standard procedure for row-reducing a matrix can be explained by my answers to these questions:

Why is getting $1$ ’s on the diagonal helpful?

Why do I want to get $0$ ’s in the non-diagonal positions of one column before I move on to working with getting a $1$ in the next column?

While not one of the “official” row-reduction steps, why might switching rows of the matrix be helpful?

Adding Notation, Vocabulary, and Conventions

The elementary row operations we can perform when row-reducing a matrix can be written in symbols instead of described in words. Translate each of the following symbolic instructions into words that describe the elementary row operation represented.

$3 R_{2} ⟶ R_{2}$

$R_{1} ⟷ R_{2}$

$3 R_{1} + R_{2} ⟶ R_{2}$

Vocabulary

elementary row operations
reduced row echelon form
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we examined a standard procedure for the row reduction of matrices. The procedure eliminated all of the guesswork of solving a system of equations with matrices, but still allowed room for sense-making and strategic thinking. We also learned notation for recording our steps in the solution process, rather than having to describe how we got one matrix from a previous one in words.

Retrieval

Create an augmented matrix that fits with the given system of equations.

1.

${\begin{cases} 5 x - 12 y = 60 \\ - 2 x + 4 = - 12 \end{cases}$

2.

${\begin{cases} - 7 x + 3 y = 8 \\ 3 x + 2 y = 6 \end{cases}$

Based on the given system of equations, determine which method for solving you would choose to use and why you think it would be the most efficient method.

3.

${\begin{cases} 6 x - 4 y = 12 \\ 2 x + 4 y = 8 \end{cases}$

4.

${\begin{cases} x = - 3 y + 4 \\ 3 x + 5 y = 9 \end{cases}$

5.

${\begin{cases} y = - 3 x + 9 \\ y = 4 x - 10 \end{cases}$

6.

${\begin{cases} x - 7 y = 2 \\ 2 x + y = 5 \end{cases}$