Properties of Determinants: Denotation Review for Matrices

1. 3.2 Properties of Determinants

2. Denotation REVIEW : the submatrix by deleting the ith row and jth column of A Example:

3. Definition For , the determinant of an matrix is REVIEW

4. REVIEW Denotation: (i, j)-cofactor of A : Theorem 1

5. Theorem 2 If A is a triangular matrix, then det A is the product of the entries on the main diagonal of A. REVIEW

6. Theorem 3 Row Operations. Let A be a square matrix. • If a mutiple of one row of A is added to another row to produce a matrix B, then det B=det A. • If two rows of A are interchanged to produce B, then det B= - det A. • If one row of A is mutiplied by k to produce B, then det B=k det A.

7. Example: Compute the determinant

8. Theorem 4.

9. Example: Find the determinant of

10. Theorem 5

11. Theorem 5 Theorem 6