Lesson 9 Solving Systems with Matrices, Revisited Solidify Understanding

Learning Focus

Use multiplicative inverse matrices to solve systems.

Is it possible to divide by a matrix? How is division defined for real numbers, and what would be the equivalent operation for matrices?

How can I use properties of matrix operations to solve matrix equations, and what does the solution to a matrix equation represent?

Open Up the Math: Launch, Explore, Discuss

When you solve linear equations, you use properties of operations, such as the associative, inverse, and identity properties.

1.

Solve the following equation for $x$ and list the properties of operations that you use during the equation-solving process.

$\frac{2}{3} x = 8$

The list of properties you used to solve this equation probably included the use of a multiplicative inverse and the multiplicative identity property. If you didn’t specifically list those properties, go back and identify where they might show up in the equation solving process for this particular equation.

Systems of linear equations can be represented with matrix equations that can be solved using the same properties that are used to solve the above equation. First, we need to recognize how a matrix equation can represent a system of linear equations.

2.

Write the linear system of equations that is represented by the following matrix equation. (Think about the procedure for multiplying matrices you developed in previous tasks.)

$[\begin{matrix} 3 & 5 \\ 2 & 4 \end{matrix}] \cdot [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} - 1 \\ 4 \end{matrix}]$

3.

Using the relationships you noticed in problem 2, write the matrix equation that represents the following system of equations.

${\begin{matrix} 2 x + 3 y = 14 \\ 3 x + 4 y = 20 \end{matrix}$

4.

The rational numbers $\frac{2}{3}$ and $\frac{3}{2}$ are multiplicative inverses. What is the multiplicative inverse of the matrix $[\begin{matrix} 2 & 3 \\ 3 & 4 \end{matrix}]$ ?

Note: The inverse matrix is usually denoted by ${[\begin{matrix} 2 & 3 \\ 3 & 4 \end{matrix}]}^{- 1}$ .

5.

The following table lists the steps you may have used to solve $\frac{2}{3} x = 8$ and asks you to apply those same steps to the matrix equation you wrote in problem 4. Complete the table using these same steps.

Original equation
$\frac{2}{3} x = 8$
Multiply both sides of the equation by the multiplicative inverse
$\frac{3}{2} \cdot \frac{2}{3} x = \frac{3}{2} \cdot 8$
Regroup the multiplication using the associative property
$(\frac{3}{2} \cdot \frac{2}{3}) x = \frac{3}{2} \cdot 8$
The product of multiplicative inverses is the multiplicative identity on the left side of the equation
$1 \cdot x = \frac{3}{2} \cdot 8$
Perform the indicated multiplication on the right side of the equation
$1 \cdot x = 12$
Apply the property of the multiplicative identity on the left side of the equation
$x = 12$

6.

What does the last line in the table in problem 5 tell you about the system of equations in problem 3?

Pause and Reflect

7.

Use the process you have just examined to solve the following system of linear equations.

${\begin{matrix} 3 x + 5 y = - 1 \\ 2 x + 4 y = 4 \end{matrix}$

Ready for More?

Here is a strategy that can be used to find the inverse of any square matrix. It is illustrated for a $3 \times 3$ matrix, but can be adapted for a matrix of any size.

Form a $3 \times 6$ matrix where the first three columns come from the matrix whose inverse is to be found, and the second three columns are the columns of the identity matrix.
Row-reduce the matrix until the first three columns are the identity matrix. The second three columns will be the inverse matrix.

Apply this strategy to find the inverse of $[\begin{matrix} 2 & 0 & 1 \\ 1 & 3 & 1 \\ 1 & 2 & 1 \end{matrix}]$

Takeaways

The system of equations $\begin{matrix} a_{1} x + b_{1} y + c_{1} z = n_{1} \\ a_{2} x + b_{2} y + c_{2} z = n_{2} \\ a_{3} x + b_{3} y + c_{3} z = n_{3} \end{matrix}$ can be represented using matrices as:

The matrix equation can be solved for $[\begin{matrix} x \\ y \\ z \end{matrix}]$ by:

The solution to the system is given by:

Lesson Summary

In this lesson, we learned a new method for solving systems of equations by representing the system with a matrix equation and using multiplicative inverse matrices to solve the equation. The solution to the matrix equation provides the solutions to the system.

Retrieval

1.

Translate triangle $E F G$ using the rule, $f (x, y) = (x + 5, y - 7)$ .

2.

Reflect triangle $E F G$ over the line $y = x + 5$ .

3.

Identify the property used in this mathematical statement.

$3 (x + 4 y) = 3 x + 12 y$

4.

Identify the property used in this mathematical statement.

$(- 7 y + 2 x) - 8 x = - 7 y + (2 x - 8 x)$