1. 2.5 Determinants and Multiplicative Inverses of Matrices Objectives: Evaluate determinants. Find inverses of matrices. Solve systems of equations by using inverse matirces.

2. Determinants Only square matrices have determinants.

3. Second-Order Determinant -40 – -77 -40 + 77 37

4. Third-Order Determinant Determinants of 3x3 matrices are called third-order determinants. One method of evaluating third-order determinants is called expansion by minors. The minor of an element is the determinant formed when the row and column containing the element are deleted.

5. Expansion of a Third-Order Determinant

6. 2(40 – 63) – 3(48 – -7) + 4(54 – -5) 2(-23) – 3(55) + 4(59) -46 – 165+ 236 25

7. Third-Order Determinant Another method for evaluating a third-order determinant is using diagonals. In this method, you begin by writing the first two columns on the right side of the determinant.

8. Diagonals Method

10. Inverse of a 2x2 Matrix A matrix must be square to have an inverse. A square matrix has an inverse if and only if its determinant is not zero.

11. Inverse of a 2x2 Matrix The scalar is the reciprocal of the determinant. a and d trade places b and c change signs

12. Step 1 – Is the matrix square? Step 2 – Find the determinant of the matrix. = (2 ∙ -3) – (-1 ∙ 1) = (-6) – (-1) = -5

13. Step 3 – Is the determinant zero? Step 4 – Fill in the formula.

14. Find the inverse of the matrix, if possible. If not possible, explain why.

15. Matrix Equation

20. Assignment 2.5 Practice Worksheet #1-8 Do #3 and 4 with minors and diagonals 2.5 pg 102 #18, 22, 23, 27, 28, 34, 38 Do #22 & 23 with both diagonals and minors