Unformatted text preview: i th row and j th column of A . So M ij is a ( k + 1) × ( k + 1) matrix with two identical rows. By our assumption, we have det M ij = 0 for all j = 1 , ··· ,k + 2 By the deﬁnition of A ij , we have A ij = (1) i + j det M ij = 0. Hence by the cofactor expansion theorem, det A = a i 1 A i 1 + a i 2 A i 2 + ··· + a i,k +2 A i,k +2 = 0 . So the statement is true for n = k + 1. ( This is the end of step two. ) By induction, det A = 0 for any ( n +1) × ( n +1) matrix with two identical rows for any n ≥ 1. / 1...
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 Spring '08
 samiabdul
 Linear Algebra, Logic, Algebra, Englishlanguage films, aij, Mathematical proof, identical rows

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