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Notes 2 Elementary Row Operations 2010

Notes 2 Elementary Row Operations 2010 - File Notes 2...

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1 File: Notes 2 Elementary Row Operations 2009.doc Elementary Row Operations (ERO) An elementary row operation on a matrix A is one of the following operations ( i A represents row i of matrix A ): (i) Row i is interchanged with row j : j i A A ←⎯→ (ii) Row i is multiplied by a nonzero scalar k : i i kA A (iii) Row i is replaced by k times row j + row i : j i i kA A A + . Elementary row operations are equivalent to pre-multiplying A by some matrix B . For example, B is obtained from an identity matrix with the following three variations for the ERO’s: (i): row (i) of B is T j e and row (j) of B is T i e ; (ii): row (i) of B is T i ke ; (iii): row (i) of B is T T j i ke e + . The solution to a system of m equations in n unknowns, Ax b = , is invariant with respect to a finite number of ERO’s. That is ˆ x solves Ax b = if and only if ˆ x solves A x b = , where ( ) , A b can be obtained from ( ) , A b by a finite number of ERO’s. Thus, there exists a matrix M such that . Ax b MAx M b A x b = = = For example, consider the matrix 2 1 1 1 2 1 1 1 2 A = . To use (i) such as interchanging rows 1 and 3, M would be
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