6.4 Determinants and Cramer s Rule

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1 6.4 Determinants and Cramer s Rule Objectives Determinant of a 2 x 2 Matrix Determinant of an 3 x 3 Matrix Determinant of a n x n Matrix Cramer s Rule 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. Determinants can be used to solve systems of linear equations, as we will see later in this section. They are also useful in determining whether a matrix has an inverse. 1

2 Determinant of a 2 2 Matrix We denote the determinant of a square matrix A by the symbol det (A) or A. We first define det (A) for the simplest cases. If A = [a] is a 1 x 1 matrix, then det (A) = a. The following box gives the definition of a 2 x 2 determinant. LOOK FAMILIAR? 2

3 Example 1 Determinant of a 2 2 Matrix Evaluate A for A = Solution: A = = 6 3 ( 3)2 = 18 ( 6) = 24 3

4 Determinant of an n n Matrix "i" is the row and "j" is the column of matrix A. 4

5 For example, if A is the matrix then the minor M 12 is the determinant of the matrix obtained by deleting the first row and second column from A. = 0(6) 4( 2) = 8 So the cofactor A 12 = ( 1) 1+2 M 12 = 8. 5

6 Find M 33 and A 33 Solution: = = 4 So A 33 = ( 1) 3+3 M 33 = 4. 6

7 Note that the cofactor of a ij is simply the minor of a ij multiplied by either 1 or 1, depending on whether i + j is even or odd. Thus in a 3 3 matrix we obtain the cofactor of any element by prefixing its minor with the sign obtained from the following checkerboard pattern. 7

8 8

9 Evaluate the determinant of the matrix. Solution using the first row: = 2( ) 3[0 6 4( 2)] [0 5 2( 2)] = = 44 9

10 In our definition of the determinant we used the cofactors of elements in the first row only. This is called expanding the determinant by the first row. In fact, we can expand the determinant by any row or column in the same way and obtain the same result in each case (although we won t prove this). The next example illustrates this principle. 10

11 Expanding by the second row, we get: Solution using the second row: = 0 + 2[2 6 ( 1)( 2)] 4[2 5 3( 2)] = = 44 11

12 Notice that we got the same determinant for this matrix no matter which row or column we use to find it. 12

13 Determinant of an 3 x 3 Matrix Short cut for finding the determinant of a 3 x 3 matrix: 13

14 Determinant of an n n Matrix The following criterion allows us to determine whether a square matrix has an inverse without actually calculating the inverse. This is one of the most important uses of the determinant in matrix algebra, and it is the reason for the name determinant. 14

Show that the matrix A has no inverse. Solution: We begin by calculating the determinant of A. Since all but one of the elements of the second row is zero, we expand the determinant by the second row.

15 Show that the matrix A has no inverse. Solution: We begin by calculating the determinant of A. Since all but one of the elements of the second row is zero, we expand the determinant by the second row. If we do this, we see from the following equation that only the cofactor A 24 will have to be calculated. = ( 0 A 21 ) + (0 A 22 ) (0 A 23 ) + (3 A 24 ) = 3A 24 = 3( 2)( ) = 0 Expand this by column 3 Since the determinant is zero, the matrix has no inverse. 15

16 Cramer s Rule Using the notation we can write the solution of the system as 16

17 Example 6 Using Cramer s Rule to Solve a System with Two Variables Use Cramer s Rule to solve the system. Solution: (-2, 1/2) 17

18 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. 18

Example 7 Using Cramer s Rule to Solve a System with Three Variables Use Cramer s Rule to solve the system. Solution: First, we evaluate the determinants that appear in Cramer s Rule.

19 Example 7 Using Cramer s Rule to Solve a System with Three Variables Use Cramer s Rule to solve the system. Solution: First, we evaluate the determinants that appear in Cramer s Rule. Note that D is the coefficient matrix and that D x, D y, and D z are obtained by replacing the first, second, and third columns of D by the constant terms. You can use either the co-factor/minor method or the short cut for each of these determinants. Solution: (39/19, 11/19, -13/38) 19

20 Cramer s Rule Cramer s Rule gives us an efficient way to solve systems of linear equations. But in systems with more than three equations, evaluating the various determinants that are involved is usually a long and tedious task (unless you are using a graphing calculator). 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. 20

21 Assignment: Section 6.4: problems 1 29 odd, odd, 61 21

Linear Algebra and Vector Analysis MATH 1120

Faculty of Engineering Mechanical Engineering Department Linear Algebra and Vector Analysis MATH 1120 : Instructor Dr. O. Philips Agboola Determinants and Cramer s Rule Determinants If a matrix is square