matrices - Finding the values of a vector if the vector.matrix product and the value of the matrix is known (only using left multiplication operations)

Given an unknown input vector $V= (v_1, v_2, v_3, v_4)$, a known $4\times 4$ matrix $A$ and a known vector-matrix product $M=[m_1,...,m_4]$. Can you discover $V$?

Normally you would just take the inverse of $A$, and right multiply it with $M$ to get $V$. However, here's the trick - in this environment, you are not allowed to right multiply, only left multiply with $4 \times 4$ matrices.

Is there a sequence of left multiplication operations that will produce the original vector? (I don't think so, but I had to ask)

(Note - no transpose operations are permitted: only $4\times 4$ multiplication)