MM1_Chapter_10

A31 a32 we observe that c1j is the determinant of the

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Unformatted text preview: erminant of a 3 × 3 matrix a1,1 a1,2 a1,3 A = a2,1 a2,2 a2,3 a3,1 a3,2 a3,3 is defined by det(A) = a1,1 a1,2 a1,3 a2,1 a2,2 a2,3 a3,1 a3,2 a3,3 = a1,1 C1,1 − a1,2 C1,2 + a1,3 C1,3 , (10.4) where C1,1 , −C1,2 , and C1,3 are the so-called cofactors of a1,1 , a1,2 , and a1,3 , respectively, and are defined by C1,1 = a2,2 a2,3 , a3,2 a3,3 C1,2 = a2,1 a2,3 , a3,1 a3,3 and C1,3 = a2,1 a2,2 . a3,1 a3,2 We observe that C1,j is the determinant of the 2 ×2 submatrix of A that is obtained by deleting the 1st row and j th column of A. Example 10.29 (determinant of a 3 × 3 Find the determinant of the matrix 10 0 1 A= 20 matrix) 1 0 . 0 10. Matrices 275 Solution: We apply formula (10.4) to compute the determinant: 101 010 200 det(A) = =1× 01 00 10 +1× −0× 20 20 00 = 1 × (1 × 0 − 0 × 0) − 0 + 1 × (0 × 0 − 1 × 2) = 1 × 0 + 1 × (−2) = −2. 2 Example 10.30 (determinant of a 3 × 3 matrix) Determine the determinant of the 3 × 3 matrix 123 A = 4 6 5 . 987 Solution: We have from (10.4) that det(A) = 123 465 987 = 1× 65 45 46 −2× +3× 87 97 98 = (6 × 7 − 5 × 8) − 2 × (4 × 7 − 5 × 9) + 3 × (4 × 8 − 6 × 9) = (42 − 40) − 2 × (28 − 45) + 3 × (32 − 54) = 2 − 2 × (−17) + 3 × (−22) = 2 + 34 − 66 = −30. 2 Remark 10.31 (use 3 × 3 matrix to determine the vector product) In Remark 9.44 we mentioned the following formula for the vector product a × b = (a1 , a2 , a3 ) × (b1 , b2 , b3 ) = ijk a1 a2 a3 b1 b2 b3 = a2 a3 a1 a3 a1 a2 i− j+ k b2 b3 b1 b3 b1 b2 = (a2 b3 − a3 b2 ) i − (a1 b3 − a3 b1 ) j + (a1 b2 − a2 b1 ) k, which we can now understand with our knowledge of the determinant notation. Now we learn the formula for the determinant of an n × n matrix. This formula is a generalization of the formula (10.4) for the determinant of a 3 × 3 matrix. 276 10. Matrices Definition 10.32 (determinant of an n × n matrix) Let A = (ai,j ) be an n × n matrix whose entry in ith row and j th column is denoted by ai,j . Then the determinant of the n × n matrix A is given by n ai,j (−1)i+j Ci,j , det(A) = (10.5) j =1 for any i ∈ {1, 2, . . . , n}, where Ci,j is the determinant of the (n − 1) × (n − 1) matrix obtained by deleting the ith row and j th column from A. The term (−1)i+j Ci,j is called the cofactor of ai,j . Remark 10.33 (comments on the determinant) (a) The notion ‘for any i ∈ {1, 2, . . . , n}’ in Definition 10.32 implies that for any choice of i the value of the determinant det(A) is the same. (b) In the above definition, we have ‘expanded about the ith row’. We can also ‘expand about the j th column’. Indeed, the determinant of A is also given by n ai,j (−1)i+j Ci,j , det(A) = (10.6) i=1 for any j ∈ {1, 2, . . . , n}. (c) For a 3 × 3 matrix a1,1 a1,2 a1,3 A = a2,1 a2,2 a2,3 , a3,1 a3,2 a3,3 expanding about the first row (that is, taking i = 1 in Definition 10.32) gives that the determinant of A is 3 a1,j (−1)1+j C1,j det(A) = j =1 = a1,1 (−1)1+1 C1,1 + a1,2 (−1)1+2 C1,2 + a1,3 (−1)1+3 C1,3 = a1,1 C1,1 − a1,2 C1,2 + a1,3 C1,3 , where C1,1 = a22 a23 , a32 a33 C1,2 = a21 a23 , a31 a33 and C1,3 = a21 a22 . a31 a32 This is the formula (10.4) given in Definition 10.28, and we see that Definition 10.28 is just a special case of Definition 10.32. 10. Matrices 277 We now come b...
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This document was uploaded on 01/31/2014.

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