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4.5 2x2 Matrices, Determinants and Inverses

Learn how to evaluate determinants, find inverses, and solve equations using 2x2 matrices. Understand the concept of multiplicative inverses and use identity matrices for verification. Discover when an inverse exists and how to calculate it efficiently.

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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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