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?

Question

zarkon · Answer

elementary matrices are invertable

anonymous · Answer

\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*}