Determinants. Cramer’s Rule

Determinants. Cramer’s Rule

1. 2018 Lecture №1 Determinants Dobroshtan О lena Cramer’s Rule

2. Determinants and Cramer’s Rule

3. Determinants If a matrix is square (that is, if it has the same number of rows as columns), then we can assign to it a number called its determinant.

4. Determinants of 2 × 2 Matrices Associated with every square matrix is a real number called the determinant of A . In this text, we use det A . The determinant of a 2 × 2 matrix A , is defined as

5. Determinant of a 2 x 2 Matrix

6. If A = Determinant of a 2  2 Matrix

7. Matrices are enclosed with round brackets, while determinants are denoted with vertical bars . A matrix is an array of numbers , but its determinant is a single number .

8. Determinants The arrows in the following diagram will remind you which products to find when evaluating a 2  2 determinant.

9. Example 1 EVALUATING A 2  2 DETERMINANT Let A = Find  A . Solution Use the definition with a 11 a 22 a 21 a 12

10. Determinant of a 2 x 2 Matrix Evaluate | A | for

11. Determinant of a 3  3 Matrix If A =

12. Determinant of an n x n Matrix

13. Cofactors The determinant of each 2  2 matrix above is called the minor of the associated element in the 3  3 matrix. The symbol represents M ij , the minor that results when row i and column j are eliminated .

14. Element Minor Element Minor a 11 a 22 a 2 1 a 23 a 3 1 a 33 Cofactors

15. The minor M 12 is the determinant of the matrix obtained by deleting the first row and second column from A . For example , A is the matrix

16. Let M ij be the minor for element a ij in an n  n matrix. The cofactor of a ij , written as A ij , is Cofactor

17. Example FINDING COFACTORS OF ELEMENTS Find the cofactor of each of the following elements of the matrix a. 6 Solution Since 6 is in the first row, first column of the matrix, i = 1 and j = 1 so The cofactor is

18. Example FINDING COFACTORS OF ELEMENTS Find the cofactor of each of the following elements of the matrix Solution The cofactor is b. 3 Here i = 2 and j = 3 so,

19. Example FINDING COFACTORS OF ELEMENTS Find the cofactor of each of the following elements of the matrix Solution The cofactor is c. 8 We have, i = 2 and j = 1 so,

20. Determinant of an n x n Matrix Similarly, So, A 33 = (–1) 3+3 M 33 = 4

21. Finding the Determinant of a Matrix Multiply each element in any row or column of the matrix by its cofactor. The sum of these products gives the value of the determinant.

22. Example EVALUATING A 3  3 DETERMINANT Evaluate expanding by the second column . Solution Use parentheses, & keep track of all negative signs to avoid errors.

23. Example EVALUATING A 3  3 DETERMINANT Evaluate expanding by the second column . Solution Now find the cofactor of each element of these minors.

24. Example EVALUATING A 3  3 DETERMINANT Evaluate expanding by the second column . Solution Find the determinant by multiplying each cofactor by its corresponding element in the matrix and finding the sum of these products.

25. Example EVALUATING A 3  3 DETERMINANT

26. Example EVALUATING A 3  3 DETERMINANT Expanding by the second row, we get:

27. Example EVALUATING A 3  3 DETERMINANT Expanding by the third column, we get:

28. Cramer’s Rule

29. Linear Equations and Determinants The solutions of linear equations can sometimes be expressed using determinants. To illustrate, let’s solve the following pair of linear equations for the variable x .

30. Cramer’s Rule Using the notation the solution of the system can be written as:

31. Cramer’s Rule for a System with Two Variables Use Cramer’s Rule to solve the system.

32. Cramer’s Rule for a System with Two Variables For this system, we have:

33. Cramer’s Rule for a System with Two Variables The solution is:

34. Cramer’s Rule Cramer’s Rule can be extended to apply to any system of n linear equations in n variables in which the determinant of the coefficient matrix is not zero.

35. As we saw in the preceding section, any such system can be written in matrix form as:

36. Caution As indicated in the preceding box, Cramer’s rule does not apply if D = 0. When D = 0 the system is inconsistent or has infinitely many solutions. For this reason, evaluate D first.

37. Example APPLYING CRAMER’S RULE TO A 2  2 SYSTEM Use Cramer’s rule to solve the system Solution By Cramer’s rule, and Find Δ first, since if Δ = 0, Cramer’s rule does not apply. If Δ ≠ 0, then find Δ x and Δ y .

38. Example APPLYING CRAMER’S RULE TO A 2  2 SYSTEM By Cramer’s rule,

39. Let an n  n system have linear equations of the form Define D as the determinant of the n  n matrix of all coefficients of the variables. Define D x 1 as the determinant obtained from D by replacing the entries in column 1 of D with the constants of the system. Define D xi as the determinant obtained from D by replacing the entries in column i with the constants of the system. If D  0, the unique solution of the system is General form of Cramer’s Rule

40. Example APPLYING CRAMER’S RULE TO A 3  3 SYSTEM Use Cramer’s rule to solve the system. Solution Rewrite each equation in the form ax + by + cz + = k .

41. Example APPLYING CRAMER’S RULE TO A 3  3 SYSTEM Verify that the required determinants are

42. Example APPLYING CRAMER’S RULE TO A 3  3 SYSTEM Thus, so the solution set is

43. Example SHOWING THAT CRAMER’S RULE DOES NOT APPLY Show that Cramer’s rule does not apply to the following system. Solution We need to show that Δ = 0. Expanding about column 1 gives Since D = 0, Cramer’s rule does not apply.

44. Note When D = , the system is either inconsistent or has infinitely many solutions.

45. Use Cramer’s Rule to solve the system. First, we evaluate the determinants that appear in Cramer’s Rule.

46. Now, we use Cramer’s Rule to get the solution:

47. Limitations of Cramer’s Rule However, in systems with more than three equations, evaluating the various determinants involved is usually a long and tedious task. Moreover, the rule doesn’t apply if | D | = 0 or if D is not a square matrix. So, Cramer’s Rule is a useful alternative to Gaussian elimination—but only in some situations.