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Kenneth KnowlesDr. Alan MorrisKilgore College

Fall 2012

ELEMENTARY LINEAR ALGEBRA

CHAPTER THREE - Determinants Section 3.1 - The Determinant of a Matrix Definition: Every square matrix can be associated with a real number called its determinant.

The determinant of a 1 by 1 matrix is the matrix entry. I.e., if then . The determinant is a scalar, not a matrix.

The determinant of a 2 by 2 matrix if

determinant

Example:

determinant

14 Definition: If A is a square matrix, then an element of the minor matrix, , is the determinant ofthe matrix obtained from by deleting the th row and th column from A. Definition: The cofactor, , of an element , is given by

.

Example: Find the minors and cofactors of

The minor matrix and cofactor matrix are then

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and .

Note the sign pattern relation between the minor and cofactor matrices:

Definition: If is a square matrix of order , then .

Example:

determinant

2det(A) =

2 Theorem 3.4: Expansion by Cofactors Let be a square matrix of order . The determinant of is given by

( th row expansion)

or ( th row expansion).

Note: The rown and column chosen to find is arbitrary.

Section 3.2 - Evaluation of a Determinant Using Elementary Operations Definition: Theorem 3.3: Elementary Row Operations and Determinants Let and be square matrices. 1) If is obtained from by interchanging two rows, then . 2) If is obtained from by adding a multiple of a row of to another row of , then

. 3) If is obtained from by multiplying a row of by a nonzero constant, c, then

.

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Theorem 3.4: Conditions That Yield a Zero Determinant If is a square matrix and one of the following conditions is true, then . 1) An entire row or an entire column consists of zeros 2) Two rows or two columns are equal 3) One row or column is a multiple of another row or column

Section 3.3 - Properties of Determinants Theorem 3.5: Determinant of a Matrix Product If are square matrices of order then . Theorem 3.6: Determinant of a Scalar Multiple of a Matrix

If is a square matrix size , and is a nonzero scalar, then , because is obtained by , thus there are factors of in the determinant. Example: Problem 4 pg 145 a)

determinant

0 b)

determinant

determinant

0 Example:

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determinant

determinant

This example illustrates that if c=5, then .

Theorem 3.7: Determinant of an Invertible Matrix A square matrix A is invertible if and only if Theorem 3.8: Determinant of an Inverse Matrix

If A is invertible, then .

Theorem 3.9: Determinant of a Transpose

If A is a square matrix, then .

Section 3.4 - Introduction to Eigenvalues Definition:

Section 3.5 - Applications of Determinants Definition: The adjoint of a matrix A is the transpose of the matrix where each element is the

cofactor of . In other words, adj , where is the cofactor matrix of . Theorem 3.10: The Inverse of a Matrix Given by Its Adjoint

If A is an invertible matrix, then .

Example:

determinant

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transpose

This example illustrates that .

Theorem 3.11: Cramer's Rule If a solution of n linear eqns in n variables has a coefficient matrix where , then the

solution of the system is given by , where the th column of is the

column of constants in the system of eqns. Example:

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determinant

(This should be )

determinant

(This should be )

determinant

(This should be )

determinant

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