Is there a matrix, $X$, that can be solved-for here?
$ \left( \begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array}\right) X = \left( \begin{array}{ccc} 0 & b & c \\ d & e & 0 \\ 0 & h & i \end{array}\right) $
I want a given $3\times3$ matrix to be transformed into the one above, with $a,f,$ and $g$ eliminated.
If $a,b,\ldots,i$ are nice so the LHS is invertible, set $$ X = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}^{-1} \begin{pmatrix} 0 & b & c \\ d & e & 0 \\ 0 & h & i \end{pmatrix} $$