1. Chapter 2 Determinants

2. With each square matrix it is possible to associate a real number called the determinant of the matrix. The value of this number will tell us whether the matrix is singular.

3. Case 1: 1×1 Matrices If A=(a) is a 1×1 matrix, then A will have a multiplicative inverse if and only if a ≠0. Thus, if we define det(A)=a Then A will be nonsingular if and only if det(A) ≠0. Case 2 : 2×2 Matrices Let then 1The Determinant of A Matrix

4. Case 3 3×3 Matrices

5. Definition Let A =( a ij ) be an nxn matrix and let M ij denote the ( n -1)x( n -1) matrix obtained from A by deleting the row and column containing a ij. The determinant of M ij is called the minor of a ij. We define the cofactor A ij of a ij by

6. Example If, then calculate det(A).

7. Definition The determinant of an nxn matrix A, denoted det(A), is a scalar associated with the matrix A that is defined inductively as follows: are the cofactors associated with the entries in the first row of A. where

8. Theorem 2.1.1 If A is an nxn matrix with n≥2, then det(A) can be expressed as a cofactor expansion using any row or column of A.

9. Example Evaluate

10. Theorem 2.1.2 If A is an nxn matrix, then det(A T )=det(A). Theorem 2.1.3 If A is an nxn triangular matrix, the determinant of A equals the product of the diagonal elements of A. Theorem 2.1.4 Let A be an nxn matrix, (1) If A has a row or column consisting entirely of zeros, then det( A )=0. (2) If A has two identical rows or two identical columns, then det( A )=0.

11. 2 Properties of Determinants Lemma 2.2.1 Let A be an nxn matrix. If A jk denotes the cofactor of a jk for k=1, …, n, then (1)

12. Effects of row operation on the the value of a determinant Row Operation I ( Two rows are interchanged.) Suppose that E is an elementary matrix of type I, then

13. Effects of row operation on the the value of a determinant Row Operation II (A row of A is multiplied by a nonzero constant.) Let E denote the elementary matrix of type II formed from I by multiplying the ith row by the nonzero constant.

14. Row Operation III (A multiple of one row is added to another row.) Let E be the elementary matrix of type III formed from I by adding c times the ith row to the jth row. Effects of row operation on the the value of a determinant

15. Ⅰ. Interchanging two rows (or columns) of a matrix changes the sign of the determinant. Ⅱ. Multiplying a single row or column of a matrix by a scalar has the effect of multiplying the value of the determinant by that scalar. Ⅲ. Adding a multiple of one row (or column) to another does not change the value of the determinant.

16. Example Evaluate

17. Main Results Theorem 2.2.2 An n×n matrix A is singular if and only if det(A)=0 Theorem 2.2.3 If A and B are n×n matrices, then det(AB)=det(A)det(B)

18. Definition The Adjoint of a Matrix Let A be an n×n matrix. We define a new matrix called the adjoint of A by 3 Cramer’s Rule

19. Example Let Compute adj A and A -1.

20. Theorem 2.3.1 (Cramer’s Rule) Let A be an nxn nonsingular matrix, and let b ∈ R n. Let A i be the matrix obtained by replacing the ith column of A by b. If x is the unique solution to Ax=b, then

21. Example Use Cramer’s rule to solve