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2x2 Matrices, Determinants and Inverses

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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2x2 Matrices, Determinants and Inverses

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  1. 2x2 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 for multiplication

  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. A Matrix with a determinant of zero is called a SINGULAR matrix

  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… Take the product of the leading diagonal, and subtract the product of the non-leading 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? • Given , the inverse of A is given by: Equation to find an inverse matrix This is called the adjoint matrix. It is formed by interchanging elements in the leading diagonal and negating elements in the non-leading diagonal

  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: Find the adjoint matrix. i.e

  27. Evaluating Determinants of 2x2 Matrices • Example 1: • Determine whether the matrix has an inverse. If an inverse exists, find it. Step 2: Find the adjoint matrix. i.e 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: Find the adjoint matrix. i.e Change signs Adjoint of M

  29. Evaluating Determinants of 2x2 Matrices • Example 1: • Determine whether the matrix has an inverse. If an inverse exists, find it. Step 2: Find the adjoint matrix. i.e Change positions Adjoint of M

  30. Evaluating Determinants of 2x2 Matrices • Example 1: • Determine whether the matrix has an inverse. If an inverse exists, find it. Step 2: Find the adjoint matrix. i.e Change positions Adjoint of M

  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.

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