# High School: Algebra

### Reasoning with Equations and Inequalities HSA-REI.C.9

9. Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

This is where actual calculations take place. We've finally gotten to the fun stuff. Your students may think otherwise, but we know the truth.

Students should already know how to form a matrix equation of AX = B from a system of linear equations and be familiar with the concept of inverse matrices. They will use the inverse of a matrix in order to solve for the variables of the matrix that is formed from given equations.

Students should know that if the determinant of a square matrix is zero (ad – bc = 0), there is no inverse to the matrix. A matrix that takes the form

has an inverse of

Assuming we've already translated our equations -2x + y = 5 and -3x + y = -2 into matrix equation form, we can find the inverse matrix of the A matrix (the one with all the coefficients).

First, we take our matrix and find its determinant. That would be -2 × 1 – 1 × -3 = -2 + 3 = 1. The inverse matrix exists. Now, let's give our creation life! After switching around the numbers, we have to multiply by the inverse of the determinant, which is 11 or just 1. So our inverse matrix is fine as is.

So your students have found the inverse matrix. Now what? Well, the whole point of this process is to find the values of x and y. (They should write that down. That's important.) To do that, we can multiply the inverse matrix we found by the B matrix (the one with all the solutions). That should give us the values for x and y.

That gives us 1 × 5 + (-1 × -2) = 5 + 2 = 7 = x, and 3 × 5 + (-2 × -2) = 15 + 4 = 19 = y. If we plug those values back into the original linear equations, they should all hold up.

While this method might seem a bit cumbersome, students should appreciate that it's especially useful for systems of equations with many, many variables. On the other hand, the inverses for matrices larger than 2 × 2 should be calculated using technology. The whole point, though, is that they can help us solve for the values of the variables.

#### Drills

1. Solve the system of equations 32x + 14y = 4 and 32x – 10y – 3 = 0 using matrices.

x = 0.11, y = 0.042

The first step is to set up our matrix equation in the proper format. It should look something like this: . We can calculate the determinant to be (-10 × 32) – (14 × 32) = -768. Using it, we can find the inverse matrix and multiply it by the B matrix: . If we multiply them properly, we should end up with x = 0.11 and y = 0.042.

2. Given the system of equations 16x + 19y = 40 and 45x + 4y = 57, what are x and y equal to? Solve using matrices.

x = 1.167, y = 1.123

We can set up our matrices as . The determinant is 4 × 16 – 45 × 19 = -791, meaning that the inverse matrix multiplied by the solutions matrix is . If we carry out this operation, we end up with x = 1.167 and y = 1.123.

3. Given the equations 67x + 14y = 13 and 87x + 9y = 17, solve for x and y.

x = 0.197, y = -0.013

We can set up our matrices to be . The determinant of the coefficient matrix is 9 × 67 – 87 × 14 = -615. That means our inverse matrix multiplied by the solutions matrix is . Multiplying the two should give us x = 0.197 and y = -0.013.

4. You're determined to find the determinant of the following matrix. Being so determined, you're never going to give up, ever, until you find it on your own! That's just your personality, or at least we'll pretend it is. Estimate in minutes, how long it'll take you to find the determinant of the following matrix.

Forever

No, we aren't insulting you. The determinant can only be defined for square matrices. This matrix isn't square, so it will take you forever to do so. Sorry if that took you forever to figure out.

5. You're really out to prove yourself (especially if you got that last question wrong). However, this time around, you and a friend are out to see who will find the inverse of your specific matrix first. Which of you will find the inverse matrix first?

Neither will find one

Since we need the determinant to find the inverse, we cannot find the inverse of a non-square matrix (at least for the purposes of our discussion). There are exceptions to this rule, but they don't apply here for now. We're really sorry if that took your forever to figure out as well.

6. What is the inverse matrix of the coefficients of the two equations 15x + 15y = 3 and 14x + 14y = 7?

There is no inverse matrix

We shouldn't even have to draw out a matrix. By finding that our determinant 15 × 14 – 15 × 14 is equal to zero, we know there cannot be an inverse matrix. Yes, the matrix is square and no, it doesn't have an inverse. No, we aren't lying and yes, we are evil.

7. Using computer technology to help you solve the system of equations using matrices, find the values of x, y, and z respectively.

14x + 19y + 13z = 415

67x + 87y + 15z = 617

88x + 87y + 90z = 561

x = -85.7, y = 69.1, z = 23.2

This looks more brutal than gluing your hand to a charging rhino, but it's really not that bad. The computer will do most of the work for us; all we really need to do is set up the matrices properly, like this:

8. The determinant of a matrix is calculated by using which equation?

All other answers are not the equation for a determinant. If you haven't memorized it (learning is more important than memorizing, anyway), you can draw a generic 2 × 2 square matrix and label its numbers  The determinant is the product of the top left and bottom right minus the product of the top right and bottom left, or adbc.

9. The inverse of a square 2 × 2 matrix is found by using which of the following?

By definition, this is how our inverse matrix is set-up. We're sorry if you got it wrong, but the only way to get that one right without answering (C) is to rewrite the definition. If you really want to do so, you can.

10. If the determinant of a matrix is found to be less than 0, then we can conclude that:

The inverse matrix exists