MM1_Chapter_10

Hence deta 4 c21 6 c22 5 c23 4 10 6 20

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Unformatted text preview: ack to the matrix from Example 10.30, and we compute its determinant in two different ways. Example 10.34 (determinant of 3 × 3 matrix) Consider again the matrix A from Example 10.30, given by 123 A = 4 6 5 . 987 (a) Expanding about the second row (that is, taking (10.5) with i = 2) gives that the determinant of A is 3 a2,j (−1)2+j C2,j det(A) = j =1 = a2,1 (−1)2+1 C2,1 + a2,2 (−1)2+2 C2,2 + a2,3 (−1)2+3 C2,3 = −a2,1 C2,1 + a2,2 C2 2 − a2,3 C2,3 = −4 C2,1 + 6 C2,2 − 5 C2,3 , where C2,1 = 23 87 = 14 − 24 = −10, C2,2 = 13 97 = 7 − 27 = −20, C2,3 = 12 98 = 8 − 18 = −10. Hence det(A) = −4 C2,1 + 6 C2,2 − 5 C2,3 = −4 × (−10) + 6 × (−20) − 5 × (−10) = 40 − 120 + 50 = −30, agreeing with the answer we obtained previously in Example 10.30. (b) Expanding about the first column (that is, taking (10.6) with j = 1) gives that the determinant of A is 3 ai,1 (−1)i+1 Ci,1 det(A) = i=1 278 10. Matrices = a1,1 (−1)1+1 C1,1 + a2,1 (−1)2+1 C2,1 + a3,1 (−1)3+1 C3,1 = a1,1 C11 − a2,1 C2,1 + a3,1 C3,1 = C1,1 − 4 C2,1 + 9 C3,1, where C1,1 = 65 87 = 42 − 40 = 2, C2,1 = 23 87 = 14 − 24 = −10, C3,1 = 23 65 = 10 − 18 = −8. Hence det(A) = C1,1 − 4 C2,1 + 9 C3,1 = 2 − 4 × (−10) + 9 × (−8) = 2 + 40 − 72 = −30, again agreeing with the answer we obtained previously in Example 10.30. 2 In the next lemma we get information about how the determinant changes if we perform certain operations on the matrix, such as interchanging rows (or columns) of the matrix, multiplying a row (or column) with a scalar, or even adding a multiple of one row (or column) to another row (or column). Such operations can be used to modify the the matrix (whose determinant we want to compute) and thus may allow a much easier computation of the determinant of the modified matrix. Lemma 10.35 (properties of the determinant) We can use the following properties of the determinant when finding the determinant of an n × n matrix: (i) multiplying all the elements of a single row or column of a matrix by a real number λ results in the determinant of the matrix being multiplied by λ; (ii) interchanging any two rows or any two columns of a matrix changes the sign of the determinant of the matrix; (iii) adding a multiple of one row (or column) of a matrix to another row (or column) does not change the determinant of the matrix; (iv) if any row or column of a matrix consists entirely of zeros, then the determinant of the matrix is zero; (v) if any two rows or any two columns of a matrix are the same, then the determinant of the matrix is zero. 10. Matrices 279 Example 10.36 (determinant of a 4 × 4 Find the determinant of the matrix 4 3 7 9 A= 0 4 3 −1 matrix) 0 2 2 4 1 3 1 0 . Solution: Rule (i) allows us to take out the common factor 2 from column 3: 4 3 9 7 det(A) = 2 × 4 0 3 −1 0 1 1 2 Next, we use rule (ii) to interchange columns 1 and left corner 1: 1 3 3 7 det(A) = 2 × (−1) × 1 0 0 −1 1 3 . 1 0 4 to make the entry in the top 0 1 1 2 4 9 . 4 3 We now apply rule (iii) twice. Firstly, we subtract three times row 1 from row 2 new (that is, r2 = r2 − 3 r1 ): 1 3 0 −2 det(A) = −2 × 1 0 0 −1 0 4 1 −3 . 1 4 2 3 new Now, we subt...
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This document was uploaded on 01/31/2014.

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