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College Algebra Exam Review 181

College Algebra Exam Review 181 - 3.5 LINEAR ALGEBRA OVER Z...

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3.5. LINEAR ALGEBRA OVER Z 191 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 ˛ . Write ˇ D ˛q C r where 0 < r < j ˛ j . A row operation of type 1, a i ! a i qa 1 produces a matrix with r in the .i; 1/ position. Then transposing rows 1 and i yields a matrix with r in the .1; 1/ position. The case that A has an entry in the first row that is not divisible by ˛ is handled similarly, with column operations rather than row operations. If ˛ divides all the entries in the first row and column, then row and column operations of type 1 can be used to replace the nonzero entries by zeros. n Proof of Proposition 3.5.9 . If A is the zero matrix, there is nothing to do. Otherwise, we proceed as follows: Step 1. There is a nonzero entry of minimum size. By row and column permutations, we can put this entry of minimum size in the .1; 1/ position. Denote the .1; 1/ entry of the matrix by ˛ . According to Lemma 3.5.10 ,
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