1. 3.1 Introduction to Determinants

2. 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?

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

4. Definition: Determinant
For , the determinant of an matrix is

5. Example: Find the determinant of the following matrix.

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

7. 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.

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

9. Example: Compute det A for

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