= Solution
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An <elementary row operation> is one of three operations: interchange two rows, multiply one row by a nonzero <scalar>, or add a scalar multiple of one row to a different row. Each operation is reversed by another operation of the same kind.
Performing such an operation on an $m\times n$ <matrix> $A$ is equivalent to left multiplication by the $m\times m$ <elementary matrix> obtained by applying that operation to the <identity matrix>. Each <elementary matrix> is invertible, with inverse corresponding to the reverse operation. Thus a sequence of operations gives
$$
B=E_kE_{k-1}\cdots E_1A,
\qquad
\boxed{B=PA,\quad P=E_k\cdots E_1\text{ invertible}.}
$$
For square <matrices>, if $A$ were invertible, then $B=PA$ would have inverse $A^{-1}P^{-1}$. Hence \b[if $B$ is singular, $A$ must be singular as well]. In fact the two are invertible together, because $A=P^{-1}B$ gives the converse.
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