Let \(A\in M_{m\times n}(F)\) and \(b\in F^m\). Show that applying the same sequence of elementary row operations to \(A\) and \(b\) does not change the solution set of \[ Ax=b. \]
Solution
Let \[ E=E_k\cdots E_1 \] be a product of elementary matrices, and set \[ A'=EA,\qquad b'=Eb. \] Every elementary matrix is invertible, hence so is \(E\).
For any \(x\in F^n\), \[ Ax=b \iff EAx=Eb \iff A'x=b'. \] The reverse implication uses multiplication by \(E^{-1}\).
Thus \(Ax=b\) and \(A'x=b'\) have exactly the same solutions.