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Section 4.5

Section 4.5. 2 x 2 Matrices, Determinants, and Inverses. Evaluating Determinants of 2 x 2 Matrices. Definition 1: A square matrix is a matrix with the same number of columns and rows.

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Section 4.5

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  1. Section 4.5 2 x 2 Matrices, Determinants, and Inverses

  2. Evaluating Determinants of 2 x 2 Matrices • Definition 1: A square matrix is a matrix with the same number of columns and rows. • Definition 2: For an nxn square matrix, the multiplicative identity matrix is an nxn square matrix I, or In, with 1’s along the main diagonal and 0’s elsewhere.

  3. Identity Matrix

  4. Evaluating Determinants of 2 x 2 Matrices • Definition 3: If A and X are n x n matrices, and AX = XA = I, then X is the multiplicative inverse of A, written A-1.

  5. Example 1 • Show that the matrices are multiplicative inverses.

  6. Example 2 • Show that the matrices are multiplicative inverses.

  7. Determinant of a 2 x 2 Matrix • Definition 4: The determinant of a 2 x 2 matrix is ad – bc.

  8. Symbols for the Determinant detA = = ad - bc

  9. Example 3 • Evaluate each determinant.

  10. Example 4 • Evaluate each determinant.

  11. Example 5 • Evaluate each determinant.

  12. TOTD • Evaluate the determinant. • Does this matrix have an inverse?

  13. Property: Inverse of a 2 x 2 Matrix • Let . If det A = 0, then A has no inverse. • If det A ≠ 0, then

  14. Example 6 • Determine whether each matrix has an inverse. If an inverse matrix exists, find it.

  15. Example 7 • Determine whether each matrix has an inverse. If an inverse matrix exists, find it.

  16. TOTD • Determine whether each matrix has an inverse. If an inverse matrix exists, find it.

  17. Quiz 4.1-4.3 Review

  18. 4.5 Review • Determinant = detA = = ad – bc • If detA 0, then: OR… in calculator: [A]-1

  19. Using Inverse Matrices to Solve Equations AX = B A-1(AX) = A-1B (A-1A)X = A-1B IX = A-1B X = A-1B

  20. Example 8 • Solve each matrix equation in the form AX = B.

  21. Example 9 • Solve each matrix equation in the form AX = B.

  22. Example 10 • Communications The diagram shows the trends in cell phone ownership over four consecutive years. • Write a matrix to represent the changes in cell phone use. • In a stable population of 16,000 people, 9927 own cell phones, while 6073 do not. Assume the trends continue. Predict the number of people who will own cell phones next year. • Use the inverse of the matrix from part (a) to find the number of people who owned cell phones last year.

  23. TOTD • Solve the matrix equation in the form of AX=B.

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