Elementary Matrices and Permutation Matrices
Let A be the m x n matrix in (I.14). An elementary m x m matrix E is a matrix such that the effect of EA is the addition of a multiple of one row of A to another row of A. For example, let Ei, j (c) be an elementary matrix such that the effect
of E,, j(c)A is that c times row j is added to row i < j:





Then E+j (c)6 is equal to the unit matrix Im (compare (1.18)) except that the zero in the (i, j)’s position is replaced by a nonzero constant c. In particular, if i = 1 and j = 2 in (I...
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