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Unformatted text preview: _ Un + 3vn 4vn c Vn+ 1 Vn 4 Vn 4 decreasing sequence. Un Vn 4 1 1  < 0, so Vn is a 44 n 1 (d) Ul = 0 = ~ i;rh, so the result is true for n = 1. Now assume it's true for n = k. Then Uk+Vk _ Uk ~( _1_) 2 2 + 2 Uk + 4k 1 1121111 Uk + 24k1 ="3 64k2 + 24k1 2 1 1 3 2 1 (4 3) 2 1 1 "3 6(4k 2 4k 1) ="3 6 4k 1 ="3 64k 1 which is the result for n = k + 1. By the Principle of Mathematical Induction, Un = ~ i 4}2 for all n ~ 1. 58. If A = raj is a 1 x 1 matrix, det A = a. Now suppose A is an n x n matrix whose (i, j) entry is aij' Let Mij be the (n  1) x (n  1) matrix obtained from A by deleting row i and column j. Then n detA = 2)I)i+ 1 ail detM il i=l (We have given the definition by "expansion of cofactors of the first column." Equally good definitions can be given for expansion of cofactors of any other row, or column, by making appropriate changes in the two occurrences of the subscript i1. See an appendix.)...
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This note was uploaded on 11/08/2010 for the course MATH discrete m taught by Professor Any during the Summer '10 term at FSU.
 Summer '10
 any
 Graph Theory

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