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.
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