Question

If a matrix B results from a matrix A by applying an elementary row operation, is there always an elementary row operation that can be applied to B to recover A?

Answer 1

elementary matrices are invertable

Answer 2

\begin{eqnarray*}
B &=& E_1E_2...E_nA \\
E_n^{-1}...E_2^{-1}E_1^{-1}B &=& E_n^{-1}...E_2^{-1}E_1^{-1} E_1 E_2...E_nA \\
E_n^{-1}...E_2^{-1}E_1^{-1}B &=& E_n^{-1}...E_2^{-1} I E_2...E_nA \\
E_n^{-1}...E_2^{-1}E_1^{-1}B &=& I A = A \\
\end{eqnarray*}