= Solution
{c}
An <elementary row operation> is one of three reversible changes: interchange two rows; multiply one row by a nonzero <scalar>; or add a <scalar multiple> of one row to a different row. Their inverses are, respectively, the same interchange, multiplication by the reciprocal, and addition of the negative <scalar multiple>.
Applied to an <augmented matrix>, these changes interchange <linear equations>, rescale an equation by a nonzero factor, or add a multiple of one equation to another. Every old solution satisfies the new equations. Applying the inverse <elementary row operation> gives the converse, so \b[the solution set is unchanged] at every step. Equivalently, all the operations together give
$$
[\widetilde A\mid\widetilde b]=E[A\mid b],\qquad\widetilde Ax=\widetilde b\iff EAx=Eb\iff Ax=b,
$$
where $E$ is an <invertible matrix>, a product of <elementary matrices>.
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