1. Inverse Matrices and Matrix Equations Dr. Shildneck 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 diagonal - zero’s every where else 2x2 Identity3x3 Identity

4. INVERSE MATRICES Inverse Matrices work like reciprocals. When you multiply a matrix by its inverse, you get the identity matrix. Inverse Matrices ARE COMMUTATIVE! [A][A -1 ] = [I] = [A -1 ][A]

5. Inverses of 2x2 Matrices 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”

6. Inverses of 2x2 Matrices Find the inverse of the matrix A. A= a b cd

7. A =A = a b cd 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. - - 1

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

9. Inverses of 2x2 Matrices Find the inverse of the matrix A. A= 2 1 64

10. Inverses of 2x2 Matrices Find the inverse of the matrix A. B= 2 1 64

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

12. Using the Calculator To find determinants for matrices bigger than 3x3, do the following: 1. Enter a (square) matrix in the calculator 2. To find a determinant : - Go back to the matrix screen - Tab to “MATH” - Choose #1 - det( - Go back to the matrix screen and pick a matrix

13. Using the Calculator To find the inverses of matrices bigger than 2x2 do the following: 1. Enter a (square) matrix in the calculator 2. To find the inverse : - Go back to the matrix screen and pick a matrix - Press the [x -1 ] button - Press [ENTER] - if you get decimals, press [MATH] [1][ENTER]

14. Using the Calculator Find each of the following. 1.2. The inverse of -530

15. 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

16. Equations… Solve each of the following WITHOUT using DIVISION. 1. 5x = 302. 2x + 8 = 24

17. Equations… Solve each of the following MATRIX equations for X. 1. AX = B2. AX + C = B

18. Equations… Solve each of the following MATRIX equations for 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. Equations… Solve each of the following MATRIX equations for 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.

20. Equations… Solve each of the following MATRIX equations for 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.

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