3Elementary row operations and their …

3 Elementary row operations and their correspondingmatricesAs we ll see, any elementary row operation can be performed by multiplying the augmentedmatrix (A|y) on theleftby what we ll call anelementary matrix. Just so this doesn tcome as a total shock, let s look at some simple matrix operations : SupposeEAis defined, and suppose the first row ofEis (1,0,0, .. ,0). Then the firstrow ofEAisidenticalto the first row ofA. Similarly, if theithrow ofEis all zeros except for a 1 in theithslot, then theithrowof the productEAis identical to theithrow ofA. It follows that if we want tochange onlyrow i of the matrixA, we should multiplyAon the left by some matrixEwith the following property:Every rowexceptrow i should be theithrow of the corresponding identity procedure that we illustrate below is used to reduceanymatrix to echelon form (notjust augmented matrices). The way it works is simple: the elementary matricesE1, E2.

The general rule here is the following: To perform an elementary row operation on the matrix A, rst perform the operation on the corresponding identity

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