1. Warm-Up 3) Find the determinant by hand. 4) Find the determinant using your calculator. 1) Multiply. Show work. 2) Multiply. Show work.

2. Daily Quiz – you may not use your notes. 3) Find the determinant by hand. 4) Find the determinant using your calculator. 1) Multiply. Show work. 2) Multiply. Show work.

3. 4.4 Identity & Inverse Matrices

4. Inverses What does “inverse” mean to you? What is the inverse of multiplication?

5. Identity In other words, 5 * __= 5? What does “identity” mean to you? What is the multiplicative identity for the real numbers? The identity for multiplication is 1 because anything multiplied by 1 will be itself.

6. a * a -1 = 1 In other words, 5 * ___=1? What do we multiply by to get the identity? Any number multiplied by its inverse will be the identity.

7. Identity Matrix A * A -1 = I A -1 *A = I Any matrix multiplied by its inverse will be the identity matrix. 2x2 Identity Matrix 3x3 Identity Matrix

8. Identity Matrix AI= A Just like 5*1 = 5… IA= A Or

9. Ex. 1 Determine whether A and B are inverses. Find out if AB = I If so, then yes they’re inverses. If not, then no they’re not inverses. Use your calculator. YES

10. Ex. 2 Determine whether A and B are inverses. YES

11. Ex. 3 Determine whether A and B are inverses. NO

12. The Inverse of a 2x2 Matrix A -1 = As long as ad-cb =0 If ad-cd=0, then the matrix has no inverse!!!!

13. Ex. 4 Find A -1, if it exists. A -1 =

14. Ex. 5 Find A -1, if it exists. A -1 =

15. Ex. 6 Find A -1, if it exists. Does not exist, because it’s not square.

16. Ex. 7 Find A -1, if it exists.

18. Solving Systems using Matrices and Inverses

19. Suppose ax = b How do you solve for x? We cannot divide matrices, but we can multiply by the inverse. AX = B A -1 A -1 IX = A -1 B A -1 X = A -1 B Solving Matrix Equations

20. Solving a Matrix Equation Solve the matrix equation AX=B for the 2x2 matrix X

21. Step 1: find A -1 A -1 = Step 2: Multiply both sides of the equation By A -1 on the left

22. You can check the Solution by mult AX to see if you get B

23. Ex. Solve

24. Assignment