# 206 - Suppose an mn matrix A can be transformed into a row...

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1 Suppose an m × n matrix A can be transformed into a row echelon form U only with the elementary row operations of adding multiples of some rows to others. Let U = E k E k 1 E 1 A , where E p is the p th elementary matrix corresponding the p th elementary row operation. Then where 1 1 1 2 1 Example: 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 . 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 0 1 0 1 k k k E E E a a b c b c ⎤ ⎡ ⎤ ⎡ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ = ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎦ ⎣ ⎦ ⎣ ±²²³²²´ ±²²³²²´ ±²²³²²´ L = [ l ij ] is a unit ( l ii = 1 for i = 1, …, m ) lower triangular matrix ( l ij = 0 for i < j ), and l ij = c ij for i > j if in an elementary row operation c ij times of row j is added to row i . U = [ u ij ] is an upper triangular matrix ( u ij = 0 for i > j ) because it is in a row echelon form.

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