1. Inverses and Systems Section 13.6

2. Warm – up: 1. 2. 3.

3. Let’s look at another example: Identity Matrix Matrix = Matrix

4. Are the two Matrices Inverses? The product of inverse matrices is the identity matrix. Identity, therefore, INVERSE Matrices Matrix Inverse Matrix = Identity Matrix

5. Are the two Matrices Inverses? The product of inverse matrices is the identity matrix. Not the Identity, therefore, NOT INVERSE Matrices

6. Systems of Equations Using Matrices Can be solved by graphing, substitution, or elimination. OR We can solve using matrices!!!

7. Write a matrix equation. coefficients variables constants Check rules. Multiply both sides by the inverse of the coefficient matrix since we want to get the variables alone. Order of multiplying is important so that dimensions match! Simplify. Therefore

8. Write a matrix equation. coefficients variables constants Check rules. Multiply both sides by inverse of coefficient matrix. (Want to get variables alone.) Simplify. Therefore Example 2:

9. 2x + 5y + 8z = 8 3x - 2y + 4z = 26 2x + 4y + 3z = -3 6x - 4y + 8z = 52 2x + 4y + 3z = -3 8x + 11z = 49 4x + 10y + 16z = 16 15x - 10y + 20z = 130 19x + 36z = 146 -152x - 288z = -1168 152x + 209z = 931 -79z = -237 z = 3 19x + 36(3) = 146 19x = 38 x = 2 2(2) + 5y + 8(3) = 8 4 + 5y + 24 = 8 5y = -20 y = -4 8x + 11z = 49 Useful to solve systems of three equations

10. Write a matrix equation. coefficients variables constants Check rules. Multiply both sides by inverse of coefficient matrix. (Want to get variables alone.) Now, using Matrices:

11. Simplify. Therefore

12. Write a matrix equation. coefficients variables constants Check rules. Multiply both sides by inverse of coefficient matrix. (Want to get variables alone.) Example 2:

13. Simplify. Therefore

14. Write a matrix equation. Be careful to line up the variables and constants! coefficients variables constants Check rules. Multiply both sides by inverse of coefficient matrix. (Want to get variables alone.) Example 3:

15. Simplify. Therefore