1. MATRICES Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org Last Updated: October 12, 2005

2. Inverses and Identities 5x = 3 Jeff Bivin -- LZHS

3. Now with Matrices This is the Identity Matrix for 2 x 2 Matrices Jeff Bivin -- LZHS

4. Let’s look at another example Jeff Bivin -- LZHS

5. New Question What do we multiply a matrix by to get the Identity? Jeff Bivin -- LZHS

6. The Inverse of a 2x2 Matrix Jeff Bivin -- LZHS

7. The Inverse of a 2x2 Matrix Jeff Bivin -- LZHS

8. The Inverse of a 2x2 Matrix Jeff Bivin -- LZHS

9. This is our Formula! Jeff Bivin -- LZHS

10. This is our Formula! Jeff Bivin -- LZHS

11. This is our Formula! Jeff Bivin -- LZHS

12. Are the two Matrices Inverses? Jeff Bivin -- LZHS The product of inverse matrices is the identity matrix. Identity, therefore, INVERSE Matrices

13. Are the two Matrices Inverses? Jeff Bivin -- LZHS The product of inverse matrices is the identity matrix. Not the Identity, therefore, NOT INVERSE Matrices

14. Jeff Bivin -- LZHS Does the Matrix have an Inverse? Let’s review the definition of the Inverse of a 2x2 Matrix

15. The Inverse of a 2x2 Matrix Jeff Bivin -- LZHS

16. Find the determinant! Jeff Bivin -- LZHS Therefore, NO inverse! Does the Matrix have an Inverse?

17. Find the determinant! Jeff Bivin -- LZHS Therefore, an inverse exists! Does the Matrix have an Inverse?

18. Find the determinant! Jeff Bivin -- LZHS Therefore, NO inverse! Does the Matrix have an Inverse? 12312 45645 78978 159 + 267 + 348 - 753 - 861 - 942 45 + 84 + 96 - 105 - 48 - 72

19. Find the determinant! Jeff Bivin -- LZHS Does the Matrix have an Inverse? 13213 24124 34234 142 + 313 + 224 - 342 - 411 - 223 8 + 9 + 16 - 24 - 4 - 12 Therefore, an inverse exists!

20. This is our Formula! Jeff Bivin -- LZHS 3x + 2y = 7 4x - 5y = 11 Solve the system using inverse matrices

21. This is our Formula! Jeff Bivin -- LZHS 2x - 4y = 9 3x - 2y = 1 Solve the system using inverse matrices