3.5. LINEAR ALGEBRA OVERZ191Proof.Suppose thatAhas an entryˇis in the first column, in the.i; 1/position and thatˇis not divisible by˛. WriteˇD˛qCrwhere0 < r <j˛j. A row operation of type 1,ai!aiqa1produces a matrix withrin the.i; 1/position. Then transposing rows1andiyields a matrix withrin the.1; 1/position. The case thatAhas an entry in the first row that isnot divisible by˛is handled similarly, with column operations rather thanrow operations.If˛divides all the entries in the first row and column, then row andcolumn operations of type 1 can be used to replace the nonzero entries byzeros.nProof of Proposition3.5.9.IfAis the zero matrix, there is nothing todo. Otherwise, we proceed as follows:Step 1.There is a nonzero entry of minimum size. By row and columnpermutations, we can put this entry of minimum size in the.1; 1/position.Denote the.1; 1/entry of the matrix by˛. According to Lemma3.5.10,
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