2. 4.4 & 4.5 Notes

3. Remember: Identity Matrices: If the product of two matrices equal the identity matrix then they are inverses.

4. IDENTITY MATRIX PROOF a = (-3)(1) + (4)(0) = -3 b = (-3)(0) + (4)(1) = 4 c = (-2)(1) + (6)(0) = -2 d = (-2)(0) + (6)(1) = 6

5. Step 2) Switch a & d Step 3) Change the signs of b&c Step 1) Find determinant A scalar, Put under 1 Step 4) Multiply scalar

6. FIND THE INVERSE Step 1: Find Determinant A (scalar), put under 1

7. Step 2: Step 3: SWITCH 5 AND -2 Change signs of 3 and 1 Step 4: Multiply scalar Answer:

8. 1. Find the inverse Step 1 Answer = Steps 2 & 3 Steps 4: Multiply scalar

9. Solving Matrix Equations 1.Find the inverse of the matrix next to the variable 2.Multiply both sides by the inverse matrix, the inverse must be on the left side when multiplying -Check for the Identity matrix

10. Step 1: Find the Inverse Matrix First

11. ( Multiply both sides by the inverse matrix on the left ) Step 2 Multiply rows by columns Solution

12. Multiply both sides by the inverse matrix) Find the inverse first!!!! Solve the Matrix Equation

13. 1.Subtract Matrix from both sides 2.Find the inverse 3.Multiply inverse by both sides (keep it left)

14. Homework: Read section 4.4 ***Define Identity and Inverse Matrices Pgs. 227-229; 1-3, 14-32e, 54-60e

15. 4.5 Solving systems using matrices.

16. A system can be written as a single matrix equation. Linear system Matrix equation Matrix A is called the Coefficient matrix. Matrix X is called the Variable matrix Matrix B is called the Constant matrix A X = B

17. Solving for x and y Step 2: Find the inverse of the Coefficient Matrix and multiply both sides Step 1: Set up the equation in matrix form

18. Step 2: Finding the Inverse Matrix

19. Step 2: Multiply both sides by the Inverse… (2,-2)

20. USE AN INVERSE MATRIX TO SOLVE THE LINEAR SYSTEM. FIRST, BEGIN BY WRITING THE EQUATIONS IN MATRIX FORM. SECOND, YOU MUST NOW FIND THE INVERSE OF THE COEFFICIENT MATRIX.

21. FIND THE INVERSE OF THE COEFFICIENT MATRIX.

22. SOLVE THE SYSTEM BY MULTIPLYING BY THE INVERSE x = 1 AND y = 2 OR (1,2)

23. More Practice 1. 3. 2. 4. (8,4)(4,4) (-1,-5) (44/5, -26/5)

24. Homework Read section 4.5 ***Matrix of variables and Matrix of constants Pg.233-235; 1-3, 12-18e, 24-30e, 48-62e