3.2 Properties of Determinants

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

Definition For , the determinant of an matrix is REVIEW

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

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

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.

Example: Compute the determinant

Theorem 4.

Example: Find the determinant of

Theorem 5

Theorem 5 Theorem 6