1. Inverse Matrices and Matrix Equations
Coach Morris
Fall, 2015

2. IDENTITY MATRICES Identity Matrices work like the number 1.
When you multiply a matrix by its identity, you get the same matrix back.
Identity Matrices ARE COMMUTATIVE!
[A][I] = [I][A] = [A]

3. IDENTITY MATRICES
Identity Matrices are square with the following characteristics :
- 1’s down the main diagonal
- zero’s every where else
2x2 Identity
3x3 Identity

4. IDENTITY MATRICES
Example:

5. INVERSE MATRICES Inverse Matrices work like reciprocals…
When you multiply a matrix by its inverse, you get the identity matrix.
You can only find the inverse of square matrices.
If the det =0, then the matrix has no inverse

6. Inverses of 2x2 Matrices 1. Find the determinant
To find the inverse of a 2x2 Matrix do the following:
1. Find the determinant
2. Switch the “down” diagonal
3. Change the sign of the “up” diagonal
4. Multiply by “1/determinant”

7. Inverses of 2x2 Matrices
Find the inverse of the matrix A. a b A= c d

8. a b c d 1 A = -1 - - 1. Find the determinant : ad - bc
2. Switch the “down” diagonal.
3. Change the signs of the “up” diagonal.
4. Multiply by 1 over the determinant.

9. -b d 1 -c a Inverses of 2x2 Matrices A-1= Given the inverse of A is...
Det(A)
A-1= -c a
What happens when the determinant is equal to zero?

10. Inverses of 2x2 Matrices
Find the inverse of the matrix A. 2 1 A= 6 4

11. Inverses of 2x2 Matrices
Find the inverse of the matrix A. 2 1 B= 6 4

12. 12 6 -3 -1 Inverses of 2x2 Matrices C =
Find the inverse of the matrix A. 12 6
C = -3 -1

13. Matrix Equations
Solving Matrix Equations is much like solving linear equations…
1. You want to isolate the unknown matrix by…
2. Adding/Subtracting matrices as needed
3. Getting rid of the matrix multipled with the unknown matrix

14. Equations…
Solve each of the following WITHOUT using DIVISION.
1. 5x = 30
2. 2x + 8 = 24

15. Equations… 1. AX = B 2. AX + C = B
Solve each of the following MATRIX equations for X.
1. AX = B
2. AX + C = B

16. Equations… 1. Solve each of the following MATRIX equations for X.
2 3 5 −1 𝑋= 1.
1. Add/Subtract to get AX (or XA) on one side
2. Find the inverse of A (multiplier)
3. Multiply by the inverse on the appropriate side (both sides of “=“)
4. Simplify your answer for X.

17. Equations…
Solve each of the following MATRIX equations for X.
𝑋 = 2.
1. Add/Subtract to get AX (or XA) on one side
2. Find the inverse of A (multiplier)
3. Multiply by the inverse on the appropriate side (both sides of “=“)
4. Simplify your answer for X.

18. Equations… 1. Solve each of the following MATRIX equations for X.
𝑋 = 1.
1. Add/Subtract to get AX (or XA) on one side
2. Find the inverse of A (multiplier)
3. Multiply by the inverse on the appropriate side (both sides of “=“)
4. Simplify your answer for X.

19. ASSIGNMENT Assignment # 7 – Inverses of 2x2 Matrices
and Matrix Equations