1. Lesson 12 – 4 Inverses of Matrices
Pg. 624 #1–13 odd, 16, 21, 23, 31, 39, 41
Lesson 12 – 4 Inverses of Matrices
Pre-calculus
To show that two matrices are inverses.
To find the multiplicative inverse of a matrix

2. Learning Objective To show that two matrices are inverses.
To find the multiplicative inverse of a matrix

3. Inverses of Matrices Remember if 𝐴 and 𝐵 are inverses, 𝐴𝐵=𝐼 and 𝐵𝐴=𝐼
*Only square matrices can have multiplicative inverses*
1. Show that matrix B is the multiplicative inverse of matrix A.
𝐴= and 𝐵= −2 3 3 −4
−2 3 3 −4 =
𝐴𝐵=
−2 3 3 − =
𝐵𝐴=

4. Inverses of Matrices 𝑎 𝑏 𝑐 𝑑
To find the inverse of a square 2 x 2 matrix, we:
Step 1: Find the determinant
𝑎 𝑏 𝑐 𝑑
=𝑎𝑑
−𝑏𝑐
Step 2: Make some changes in your matrix:
Switch 𝑎 & 𝑑 spots
𝑑 −𝑏 −𝑐 𝑎
Change the sign of 𝑐 & 𝑏
Step 3: Multiply
1 𝑑𝑒𝑡 𝑑 −𝑏 −𝑐 𝑎

5. Inverses of Matrices 2. Find the multiplicative inverse. 𝐴= 9 4 7 3𝐴=
1 𝑑𝑒𝑡 𝑑 −𝑏 −𝑐 𝑎
𝑑𝑒𝑡
=9(3)
−4(7)
=27−28
=−1
1 −1
3 −4 −7 9
= −3 4 7 −9
𝐴 −1 = −3 4 7 −9

6. Inverses of Matrices For 3 x 3 and higher, we can use a graphing calc!
3. Find the multiplicative inverse.
𝐴= 4 −2 3 3 −5 −2 2 4 −3
MATRIX: 2nd x-1
 EDIT
 [A]
Enter data
QUIT: 2nd Mode
MATRIX: 2nd x-1
 Choose [A]
𝐴 −1 = − − −7 74
[𝐴] −1 =
Yikes!!
Change to fractions
MATH
 1: Frac
*arrow over to see the rest*

7. We can use inverses to solve for an unknown matrix
Inverses of Matrices
*Be careful of the order*
If 𝐴, 𝑋, 𝑎𝑛𝑑 𝐵 are matrices, and 𝐴𝑋=𝐵
To “get rid” of 𝐴, we multiply by 𝐴 −1
𝐴 −1 (𝐴𝑋)= 𝐴 −1 𝐵
𝑋= 𝐴 −1 𝐵

8. Inverses of Matrices 𝐴𝑋=𝐵 𝑋= 𝐴 −1 𝐵 4. Solve for X 2 −3 4 −5 𝑋= 6 11
2 −3 4 −5 𝑋= 6 11
2 −3 4 −5 −1 =
−5 3 −4 2
1 −10−(−12)
𝐴𝑋=𝐵
= −5 3 −4 2
= − −2 1
𝑋= 𝐴 −1 𝐵
= −1
𝑋= −5 3 −
= −2
= −1
𝑋= − −
= 1.5 −1

9. Inverses of Matrices A𝑋=𝐵 𝑋= 𝐴 −1 𝐵
5. Solve for X. (Use your calculator!)
Inverses of Matrices
− 𝑋= ↑
enter for matrix A ↑
enter for matrix B
A𝑋=𝐵
𝑋= 𝐴 −1 𝐵
𝑋= 𝐴 −1 𝐵
𝑋= 10 −2 8
𝑋= [𝐴] −1 [𝐵]

10. We can take a system of equations and turn it into a matrix equation to then solve!
Systems
6. Set up the matrices to solve the system
6𝑥−𝑦=22 2𝑥+𝑦=2
A𝑋=𝐵
coefficients
answers
You continue to solve like the
previous examples 
6 −1 2 1
= 22 2 𝑋
𝑋= 𝐴 −1 𝐵 ↑
represents
𝑥 𝑦
= 1 6−(−2) −
𝑋= 𝐴 −1 𝐵
= −
= 3 −4
(3, −4)

11. 12-6 Check up 6 −3 2 10 15 1. Solve for X. Show work. No calculator.
5 −3 4 −2 𝑋= 5 10
12-6 Check up
2. Solve for X. Use calculator!! −6 8 − 𝑋= −
6 −3 2
10 15

12. Assignment
Pg. 624 #1–13 odd, 16, 21, 23, 31, 39, 41