3 3 Determinants CH 3 3 Determinants 1

3. 3 Determinants CH. 3. 3 : Determinants 1

Definition (Determinant) Suppose A is an n× n matrix. Associated with A there is a number called the determinant of A, denoted by det A. A determinant of an n× n matrix is said to be a determinant of order n. CH. 3. 3 : Determinants 2

Finding the determinant value For a 1× 1 matrix A=[a] we have det A= |a| = a For example, 1 In this case the vertical bars | | around a number do not mean absolute value. The determinant of CH. 3. 3 : Determinants 3

Illustrative Example The determinant CH. 3. 3 : Determinants 4

We say that det A has been expanded by cofactors along the first row, with the cofactors of a 11, a 22 and a 33 being the determinants CH. 3. 3 : Determinants 5

In general, the cofactor of aij is the determinant where Mij is the determinant of the sub-matrix obtained by deleting the ith row and the jth column of A. The determinant Mij is called a minor determinant. A cofactor is a signed minor determinant CH. 3. 3 : Determinants 6

The sign factor 1 or -1 associated with a cofactor can be obtained from the checkerboard pattern: the determinant of the matrix A can be rearranged and factored again as CH. 3. 3 : Determinants 7

Example 7 Evaluate the determinant of Solution Using cofactor expansion along the first row CH. 3. 3 : Determinants 8

Another method det(A)=|A|= CH. 3. 3 : Determinants 9

CH. 3. 3 : Determinants 10

Example 8 Find the determinant of each matrix Solution CH. 3. 3 : Determinants 11

Exercise 10. 4 page 914 -915 19 In Exercises 1 to 8, evaluate the determinant. CH. 3. 3: Determinants 12

Note If the determinant of a matrix is zero, the matrix is called singular, otherwise it is called nonsingular Example 10 State whether each of the following matrix is singular or nonsingular CH. 3. 3 : Determinants 13

Solution CH. 3. 3 : Determinants 14

Exercise 10. 4 page 914 -915 20 In Exercises 17 to 26, evaluate the determinant by expanding by cofactors. CH. 3. 3: Determinants 15