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5.3 - 1 10 TH EDITION LIAL HORNSBY SCHNEIDER COLLEGE ALGEBRA.

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Presentation on theme: "5.3 - 1 10 TH EDITION LIAL HORNSBY SCHNEIDER COLLEGE ALGEBRA."— Presentation transcript:

1 5.3 - 1 10 TH EDITION LIAL HORNSBY SCHNEIDER COLLEGE ALGEBRA

2 5.3 - 2 5.3 Determinant Solution of Linear Equations Determinants Cofactors Evaluating n  n Determinants Cramer’s Rule

3 5.3 - 3 Determinants Every n  n matrix A is associated with a real number called the determinant of A, written  A . The determinant of a 2  2 matrix is defined as follows.

4 5.3 - 4 Determinant of a 2  2 Matrix If A =

5 5.3 - 5 Note Matrices are enclosed with square brackets, while determinants are denoted with vertical bars. A matrix is an array of numbers, but its determinant is a single number.

6 5.3 - 6 Determinants The arrows in the following diagram will remind you which products to find when evaluating a 2  2 determinant.

7 5.3 - 7 Example 1 EVALUATING A 2  2 DETERMINANT Let A = Find  A . Use the definition with Solution a 11 a 22 a 21 a 12

8 5.3 - 8 Determinant of a 3  3 Matrix If A =

9 5.3 - 9 Evaluating The terms on the right side of the equation in the definition of  A  can be rearranged to get Each quantity in parentheses represents the determinant of a 2  2 matrix that is the part of the matrix remaining when the row and column of the multiplier are eliminated, as shown in the next slide.

10 5.3 - 10 Evaluating

11 5.3 - 11 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. The following table in the next slide gives some of the minors from the previous matrix.

12 5.3 - 12 Cofactors ElementMinorElementMinor a 11 a 22 a21a21 a 23 a31a31 a 33

13 5.3 - 13 Cofactors In a 4  4 matrix, the minors are determinants of matrices. Similarly, an n  n matrix has minors that are determinants of matrices. To find the determinant of a 3  3 or larger matrix, first choose any row or column. Then the minor of each element in that row or column must be multiplied by +1 or – 1, depending on whether the sum of the row number and column number is even or odd. The product of a minor and the number +1 or – 1 is called a cofactor.

14 5.3 - 14 Cofactor 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

15 5.3 - 15 Example 2 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

16 5.3 - 16 Example 2 FINDING COFACTORS OF ELEMENTS Find the cofactor of each of the following elements of the matrix b. 3 Solution Here i = 2 and j = 3 so, The cofactor is

17 5.3 - 17 Example 2 FINDING COFACTORS OF ELEMENTS Find the cofactor of each of the following elements of the matrix c. 8 Solution We have, i = 2 and j = 1 so, The cofactor is

18 5.3 - 18 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.

19 5.3 - 19 Example 3 EVALUATING A 3  3 DETERMINANT Evaluate expanding by the second column. Solution Use parentheses, & keep track of all negative signs to avoid errors.

20 5.3 - 20 Example 3 EVALUATING A 3  3 DETERMINANT Now find the cofactor of each element of these minors.

21 5.3 - 21 Example 3 EVALUATING A 3  3 DETERMINANT Find the determinant by multiplying each cofactor by its corresponding element in the matrix and finding the sum of these products.

22 5.3 - 22 Cramer’s Rule Determinants can be used to solve a linear system in the form (1) (2) by elimination as follows. Multiply (1) by b 2. Multiply (2) by – b 1. Add.

23 5.3 - 23 Cramer’s Rule Multiply (1) by – a 2. Multiply (2) by a 1. Add. Similarly,

24 5.3 - 24 Cramer’s Rule Both numerators and the common denominator of these values for x and y can be written as determinants, since

25 5.3 - 25 Cramer’s Rule Using these determinants, the solutions for x and y become

26 5.3 - 26 Cramer’s Rule We denote the three determinants in the solution as

27 5.3 - 27 Note The elements of D are the four coefficients of the variables in the given system. The elements of D x are obtained by replacing the coefficients of x in D by the respective constants, and the elements of D y are obtained by replacing the coefficients of y in D by the respective constants.

28 5.3 - 28 Cramer’s Rule for Two Equations in Two Variables Given the system if then the system has the unique solution where

29 5.3 - 29 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.

30 5.3 - 30 Example 4 APPLYING CRAMER’S RULE TO A 2  2 SYSTEM Use Cramer’s rule to solve the system Solution By Cramer’s rule, and Find D first, since if D = 0, Cramer’s rule does not apply. If D ≠ 0, then find D x and D y.

31 5.3 - 31 Example 4 APPLYING CRAMER’S RULE TO A 2  2 SYSTEM By Cramer’s rule, The solution set is as can be verified by substituting in the given system.

32 5.3 - 32 General form of Cramer’s Rule 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 x1 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

33 5.3 - 33 Example 5 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.

34 5.3 - 34 Example 5 APPLYING CRAMER’S RULE TO A 3  3 SYSTEM Verify that the required determinants are

35 5.3 - 35 Example 5 APPLYING CRAMER’S RULE TO A 3  3 SYSTEM Thus, and so the solution set is

36 5.3 - 36 Caution As shown in Example 5, each equation in the system must be written in the form ax + by + cz +    = k before using Cramer’s rule.

37 5.3 - 37 Example 6 SHOWING THAT CRAMER’S RULE DOES NOT APPLY Show that Cramer’s rule does not apply to the following system. We need to show that D = 0. Expanding about column 1 gives Since D = 0, Cramer’s rule does not apply. Solution

38 5.3 - 38 Note When D = 0, the system is either inconsistent or has infinitely many solutions. Use the elimination method to tell which is the case. Verify that the system in Example 6 is inconsistent, so the solution set is ø.


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