3.1 Introduction to Determinants

REVIEW Recall: A 2x2 matrix is invertible iff its determinant is nonzero. If the matrix is invertible, then the inverse can be found as follows: Question: How do you find the determinant of larger square matrices?

Notation : the submatrix formed by deleting the ith row and jth column of A Example:

Definition: Determinant For , the determinant of an matrix is

Example: Find the determinant of the following matrix.

Definition: (i, j)-cofactor of A : Example:

Denotation: (i, j)-cofactor of A : Theorem 1: The determinant of an nxn matrix A can be computed by a cofactor expansion across any row or any column.

Example: Calculate the determinant using the 3rd row. Example: Calculate the determinant using the 2nd column.

Example: Compute det A for

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