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Unformatted text preview: a is strictly smaller than b , we mean that j a j < j b j . Lemma 8.4.8. Suppose that A has nonzero entry in the .1;1/ position. (a) If there is a element in the rst row or column that is not divisible by , then A is equivalent to a matrix with smaller .1;1/ entry. (b) If divides all entries in the rst row and column, then A is equivalent to a matrix with .1;1/ entry equal to and all other entries in the rst row and column equal to zero. Proof. Suppose that A has an entry is in the rst column, in the .i;1/ position and that is not divisible by . Any greatest common of and satses j j < j j , by the previous lemma. There exist s;t 2 R such that D s C t . Consider the matrix s t = = . This matrix has determinant equal to 1 , so it is invertible in Mat 2 .R/ . Notice that...
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 Fall '08
 EVERAGE
 Algebra, Matrices

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