Math225_Q3_soln

Math225_Q3_soln - Math 225 Quiz 3 Solutions 1 −2 3 0 4 0 7 −2 Use cofactor expansion to find the determinant 1 Consider the matrix A = 0 1 3 4

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Unformatted text preview: Math 225: Quiz 3 Solutions 1 −2 3 0 4 0 7 −2 . Use cofactor expansion to find the determinant. 1. Consider the matrix A = 0 1 3 4 1 5 −2 0 SOLUTION: We choose a row or column to expand by. Since the fourth column of A has the most zeros, we will expand by it. (Note you could have chosen a different row/column and get the same solution). 40 7 1 −2 3 1 −2 3 1 −2 3 3 − (4)det 4 0 7 + (0)det 4 0 7 A = −(0)det 0 1 3 + (−2)det 0 1 1 5 −2 1 5 −2 1 5 −2 013 1 −2 3 1 −2 3 0 1 − 4 · det 4 0 3 7 = −2 · det 1 5 −2 1 5 −2 = −2 1 · det 13 −2 3 −2 3 1 −2 + 1 · det − 4 − 4 · det − 7 · det (1) 5 −2 13 5 −2 15 = −2[(1)(−2) − (3)(5) + (−2)(3) − (1)(3)] − 4[−4((−2)(−2) − (3)(5)) − 7((1)(5) − (−2)(1))] = −2[−2 − 15 − 6 − 3] − 4[−4(4 − 15) − 7(5 + 2)] = −2(−26) − 4(44 − 49) = 52 + 20 = 72 In (1) we again used cofactor expansion to find the determinant of both resulting matrices; the first expanded by the first column and the second expanded by the second row. 21 . Is A invertible? 32 2. Compute the determinant of A = SOLUTION: A = (2)(2) − (3)(1) = 4 − 3 = 1. Since A = 0 we have that A is invertible. abc d e f −3b −3c . 3. Let A = d e f with det(A) = -6. Find det(B ) where B = −3a ghi g − 4d h − 4e i − 4f SOLUTION: d e f −3a −3b −3c −3b −3c = - d e f B = −3a g − 4d h − 4e i − 4f g − 4d h − 4e i − 4f abc a b c ab e f =3 d e f + d = 3 A + (−4) d e ghi −4d −4e −4f de a b c e f = -(-3) d g − 4d h − 4e i − 4f c f = 3(-6 + (-4)(0)) = -18 f 1 ...
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This note was uploaded on 03/30/2010 for the course MATH 225 taught by Professor Guralnick during the Spring '07 term at USC.

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