1File: Notes 2 Elementary Row Operations 2009.doc Elementary Row Operations (ERO) An elementary row operation on a matrix Ais one of the following operations (iArepresents row iof matrix A): (i)Row iis interchanged with row j: jiAA←⎯→(ii)Row iis multiplied by a nonzero scalar k: iikAA→(iii)Row iis replaced by ktimes row j+ row i: jiikAAA+→. Elementary row operations are equivalent to pre-multiplying Aby some matrix B. For example, Bis obtained from an identity matrix with the following three variations for the ERO’s: (i): row (i) of Bis Tjeand row (j) of Bis Tie; (ii): row (i) of Bis Tike; (iii): row (i) of Bis TTjikee+. The solution to a system of mequations in nunknowns, Axb=, is invariant with respect to a finite number of ERO’s. That is ˆxsolves Axb=if and only ifˆxsolves A xb′′=, where (),A b′′can be obtained from (),A bby a finite number of ERO’s. Thus, there exists a matrix Msuch that .AxbMAxM bA xb==′′=For example, consider the matrix 211121112A⎡⎤⎢⎥=−⎢⎥−⎢⎥⎣⎦. To use (i) such as interchanging rows 1 and 3, Mwould be
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