Unformatted text preview: if one is transformed into the other by a sequence of elementary row operations; likewise, two matrices are column–equivalent if one is transformed into the other by a sequence of elementary column operations. Two matrices are equivalent if one is transformed into the other by a sequence of elementary row and column operations. In the following discussion, when we say that a is smaller than b , we mean that j a j ² j b j ; when we say that a is strictly smaller than b , we mean that j a j < j b j . Lemma 3.5.10. 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....
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 Fall '08
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 Algebra, Multiplication, Invertible matrix, Elementary Row Operation, elementary Row, Matm .Z/

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