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hw2 - MAE 107 Problem JDG-NA02 1 As the first step in...

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MAE 107 Problem JDG-NA02 1.) As the first step in Gaussian elimination, i.e. reduction of a given m × n matrix A to row echelon form with pivoting, one normally exchanges a pivot row r T p ( A ), whose leading non-zero element has the greatest absolute value, with the first row r T 1 ( A ). This permutation serves mainly to help one keep track of operations during manual calculation or to simplify indexing in computer codes but is otherwise not necessary. To make this more evident, suppose that the row with largest element happens to be row p = 2, (since the initial numbering of rows is arbitrary). Then, the above row exchange is given by the permutation 1 P = I - e 1 e T 1 - e 2 e T 2 + e 1 e T 2 + e 2 e T 1 (1) This permutation is then followed by a elementary reduction step, consisting of multiplication of new Row 1, ˜ r T 1 r T 2 , by an appropriate constant γ i followed by addition to row i > 1, with γ i chosen to reduce the leading element of each rows i > 1 to zero.
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