UNIT 3.6 SOLVING SYSTEMSUNIT 3.6 SOLVING SYSTEMSUSING MATRICESUSING MATRICES

Warm UpMultiple the matrices.

1.

Find the determinant.

2. 3. –1 0

Determine whether a matrix has an inverse.Solve systems of equations using inverse matrices.

Objectives

The identity matrix I has 1’s on the main diagonal and 0’s everywhere else.

Remember!

Example 1A: Determining Whether Two Matrices Are Inverses by using your graphing calculator

Determine whether the two given matrices are inverses.

The product is the identity matrix I, so the matrices are inverses.

Example 1B: Determining Whether Two Matrices Are Inverses by using your graphing calculator

Determine whether the two given matrices are inverses.

Neither product is I, so the matrices are not inverses.

Check It Out! Example 1

Determine whether the given matrices are inverses.

The product is the identity matrix I, so the matrices are inverses.

Example 2: Finding the Inverse of a Matrix by using your graphing calculator

Find the inverse of the matrix if it is defined.

Check It Out! Example 3

Find the inverse of , if it is defined.

To solve systems of equations with the inverse, you first write the matrix equation AX = B, where A is the coefficient matrix, X is the variable matrix,and B is the constant matrix.

You can use the inverse of a matrix to solve a system of equations. This process is similar to solving an equation such as 5x = 20 by multiplying

each side by , the multiplicative inverse of 5.

The matrix equation representing is shown.

To solve AX = B, multiply both sides by the inverse A-1.

A-1AX = A-1B

IX = A-1B

X = A-1B

The product of A-1 and A is I.

Example 4: Solving Systems Using Inverse Matrices

Write the matrix equation for the system and solve.

Step 1 Set up the matrix equation.

Write: coefficient matrix variable matrix = constant matrix.

A X = B

Example 4 Continued

.

Step 2: Solve by plugging [A]-1 B in to your graphing calculator

Solution is

5

2

x

y

Example 5

Write the matrix equation for and solve.3 2 1

3 10

2 3 3

x y z

x y z

x y z

Lesson Quiz: Part I

yes

1. Determine whether and are

inverses.

2. Find the inverse of , if it exists.

Lesson Quiz: Part II

Write the matrix equation and solve.

3.

4. Decode using .

"Find the inverse."

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