Chapter 2

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. The Determinant of A Matrix 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

4. Case 3 3×3 Matrices

5. Definition Let A=(aij) be an nxn matrix and let Mijdenote the (n-1)x(n-1) matrix obtained from A by deleting the row and column containing aij. The determinant of Mij is called the minor of aij. We define the cofactorAijof aijby

6. Example If , thencalculate 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: where are the cofactors associated with the entries in the first row of A.

8. Theorem 2.1.1If 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. ExampleEvaluate

10. Theorem 2.1.2If A is an nxn matrix, then det(AT)=det(A). Theorem 2.1.3If A is an nxn triangular matrix, the determinant of A equals the product of the diagonal elements of A. • Theorem 2.1.4Let A be an nxn matrix, • 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 2.2.1Let A be an nxn matrix. If Ajkdenotes the cofactor of ajkfor 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. Effects of row operation on the the value of a determinant Row Operation III (A multiple of one row is added to another row.) Let E be the elementary matrix of type III formed from Iby adding c times the ith row to the jth row.

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. ExampleEvaluate

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

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

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∈Rn. Let Ai 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