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# 4.5 2x2 Matrices, Determinants and Inverses

4.5 2x2 Matrices, Determinants and Inverses. Evaluating Determinants of 2x2 Matrices Using Inverse Matrices to Solve Equations. Evaluating Determinants of 2x2 Matrices.

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## 4.5 2x2 Matrices, Determinants and Inverses

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1. 4.52x2 Matrices, Determinants and Inverses Evaluating Determinants of 2x2 Matrices Using Inverse Matrices to Solve Equations

2. Evaluating Determinants of 2x2 Matrices • When you multiply two matrices together, in the order AB orBA, and the result is the identity matrix, then matrices A and B are inverses. Identity matrix

3. Evaluating Determinants of 2x2 Matrices You only have to prove ONE of these. • To show two matrices are inverses… • AB = IORBA = I • AA-1 = IORA-1A = I Inverse of A Inverse of A

4. Evaluating Determinants of 2x2 Matrices • Example 1: • Show that B is the multiplicative inverse of A.

5. Evaluating Determinants of 2x2 Matrices • Example 1: • Show that B is the multiplicative inverse of A.

6. Evaluating Determinants of 2x2 Matrices • Example 1: • Show that B is the multiplicative inverse of A. AB = I. Therefore, B is the inverse of A and A is the inverse of B.

7. Evaluating Determinants of 2x2 Matrices • Example 1: • Show that B is the multiplicative inverse of A. AB = I. Therefore, B is the inverse of A and A is the inverse of B. Check by multiplying BA…answer should be the same

8. Evaluating Determinants of 2x2 Matrices • Example 1: • Show that B is the multiplicative inverse of A. AB = I. Therefore, B is the inverse of A and A is the inverse of B. Check by multiplying BA…answer should be the same

9. Evaluating Determinants of 2x2 Matrices • Example 2: • Show that the matrices are multiplicative inverses.

10. Evaluating Determinants of 2x2 Matrices • Example 2: • Show that the matrices are multiplicative inverses. BA = I. Therefore, B is the inverse of A and A is the inverse of B.

11. Evaluating Determinants of 2x2 Matrices • The determinant is used to tell us if an inverse exists. • If det ≠ 0, an inverse exists. • If det = 0, no inverse exists.

12. Evaluating Determinants of 2x2 Matrices • To calculate a determinant…

13. Evaluating Determinants of 2x2 Matrices • To calculate a determinant… Multiply along the diagonal

14. Evaluating Determinants of 2x2 Matrices • To calculate a determinant… Multiply along the diagonal Equation to find the determinant

15. Evaluating Determinants of 2x2 Matrices • Example 1: Evaluate the determinant.

16. Evaluating Determinants of 2x2 Matrices • Example 1: Evaluate the determinant.

17. Evaluating Determinants of 2x2 Matrices • Example 1: Evaluate the determinant.

18. Evaluating Determinants of 2x2 Matrices • Example 1: Evaluate the determinant. det = -23 Therefore, there is an inverse.

19. Evaluating Determinants of 2x2 Matrices • Example 2: Evaluate the determinant.

20. Evaluating Determinants of 2x2 Matrices • Example 2: Evaluate the determinant.

21. Evaluating Determinants of 2x2 Matrices • Example 2: Evaluate the determinant. det = 0 Therefore, there is no inverse.

22. Evaluating Determinants of 2x2 Matrices • How do you know if a matrix has an inverse ANDwhat that inverse is? Equations to find an inverse matrix p.201

23. Evaluating Determinants of 2x2 Matrices • Example 1: • Determine whether the matrix has an inverse. If an inverse exists, find it.

24. Evaluating Determinants of 2x2 Matrices • Example 1: • Determine whether the matrix has an inverse. If an inverse exists, find it. Step 1: Find det M

25. Evaluating Determinants of 2x2 Matrices • Example 1: • Determine whether the matrix has an inverse. If an inverse exists, find it. Step 1: Find det M det M = -2, the inverse of M exists.

26. Evaluating Determinants of 2x2 Matrices • Example 1: • Determine whether the matrix has an inverse. If an inverse exists, find it. Step 2: Rewrite the matrix in form.

27. Evaluating Determinants of 2x2 Matrices • Example 1: • Determine whether the matrix has an inverse. If an inverse exists, find it. Step 2: Rewrite the matrix in form. Change signs

28. Evaluating Determinants of 2x2 Matrices • Example 1: • Determine whether the matrix has an inverse. If an inverse exists, find it. Step 2: Rewrite the matrix in form. Change signs

29. Evaluating Determinants of 2x2 Matrices • Example 1: • Determine whether the matrix has an inverse. If an inverse exists, find it. Step 2: Rewrite the matrix in form. Change positions

30. Evaluating Determinants of 2x2 Matrices • Example 1: • Determine whether the matrix has an inverse. If an inverse exists, find it. Step 2: Rewrite the matrix in form. Change positions

31. Evaluating Determinants of 2x2 Matrices • Example 1: • Determine whether the matrix has an inverse. If an inverse exists, find it. Step 3: Use the equation to find the inverse.

32. Evaluating Determinants of 2x2 Matrices • Example 1: • Determine whether the matrix has an inverse. If an inverse exists, find it. Step 3: Use the equation to find the inverse.

33. Evaluating Determinants of 2x2 Matrices • Example 2: • Determine whether the matrix has an inverse. If an inverse exists, find it.

34. Evaluating Determinants of 2x2 Matrices • Example 2: • Determine whether the matrix has an inverse. If an inverse exists, find it.

35. Evaluating Determinants of 2x2 Matrices • Example 2: • Determine whether the matrix has an inverse. If an inverse exists, find it.

36. Homework • p.203 #1, 2, 4, 5, 14, 15, 27, 28, 32, 34

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