This problem outlines a proof that two linear systems LS1 and LS2 are equivalent (that is, have the same solution set ) if their augmented coefficient matrices A1 and A2 are row equivalent.
(a) If a single elementary row operation transforms A1 to A2, show directly—considering separately the three cases—that every solution of LS1 is also a solution of LS2.
(b) Explain why it now follows from Problem 1 that every solution of either system is also a solution of the other system; thus the two systems have the same solution set.
Problem 1
This problem deals with the reversibility of elementary row operations.
(a) If the elementary row operation cRp changes the matrix A to the matrix B, show that (1/c)/Rp changes B to A.
(b) If SWAP(Rp, Rq) changes A to B, show that SWAP(Rp, Rq) also changes B to A.
(c) If cRp + Rq changes A to B, show that (–c) Rp + Rq changes B to A.
(d) Conclude that if A can be transformed into B by a finite sequence of elementary row operations, then B can similarly be transformed into A.
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