6.2 Properties of Determinants

1. 6.2 Properties of Determinants

2. Finding a determinant using row reductions One can find the determinant of a matrix by performing row reductions. With the following properties: • If B is obtained from A by dividing a row of A by a scalar k then det(B) =1/k(det(A)) OR kdet(B) =det(A) 2) If B is obtained from A by a row swap then det(B) = -det(A) 3) If B is obtained from A by adding a multiple of one row to another row then det(B) = det(A)

3. Example 1 Find the determinant of the matrix by row reductions

4. Example 1 Solution

5. Problems 12 and 14

6. 12 and 14 Solution

7. Problem 11

8. Problem 11 Solution • Det(A) = 8 Det(B) = (8)(-9) = -72

9. To determine if a matrix is singular A square matrix is invertible if and only if Det(A)≠0 A square matrix is singular if and only if Det(A)=0

10. Example 3 • Determinant of a productdet(AB) = det(A)*det(B) Use this fact to show that there is no matrix such that

11. Example 3 Solution

12. Determinant of the Transpose of a Matrix Why is this true?

13. Homework p.273 1-15 all ,29

14. Determinant of the inverse of a Matrix