Section 8 - Geneva 304



Section 8.4: Guided Notes

The Determinant of a Square Matrix

I. The Determinant of a Square Matrix

• With each square matrix there corresponds a unique real number called the determinant of the matrix.

[pic]

[pic]

Example 1. Find the determinant of

[pic]

II. Minors and Cofactors

• To find the determinant of a matrix larger than order 2, we need to have minors and cofactors.

• If A is a square matrix, the minor of a[pic], denoted by M[pic], is the determinant of the square matrix of order n – 1, formed from A by deleting the ith row and the jth column.

• The cofactor of a[pic], denoted by C[pic], is

[pic]

Example 2. Find M[pic], C[pic], M[pic], C[pic], M[pic], and C[pic].

[pic]

• the [pic] factor of the cofactor alternates sign.

III. The Determinant of a Square Matrix

• If A is square matrix, then the determinant of A is the sum of the products of each element in any row (or column) and its cofactor.

Example 3. Find the value of the determinant of the following matrices.

[pic]

[pic]

[pic]

[pic]

IV. Triangular Matrices

• Definitions:

1. A square matrix is in upper triangular form if it has all zero entries below its main diagonal.

2. A square matrix is in lower triangular form if it has all zero entries above its main diagonal.

3. A square matrix is in diagonal form if it has all zero entries except, possibly, along the main diagonal.

• The determinant of a matrix in upper triangular, lower triangular, or diagonal form is the product of all the entries along the main diagonal.

Example 4. Evaluate the following determinants.

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download