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 ˇ in the ﬁrst 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 ﬁrst row and column, then A is equivalent to a matrix with .1;1/ entry equal to ˛ and all other entries in the ﬁrst row and column equal to zero. Proof. Suppose that A has an entry ˇ is in the ﬁrst column, in the .i;1/ position and that ˇ is not divisible by ˛ . Any greatest common ı of ˛ and ˇ satsﬁes j ı j < j ˛ j , by the previous lemma. There exist s;t 2 R such that ı D s˛ C tˇ . Consider the matrix ± s t ´ ˇ=ı ˛=ı ² . This matrix has determinant equal to 1 , so it is invertible in Mat 2 .R/ . Notice that...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra, Matrices

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