Suppose $Mx=z$ Where $M$ is a matrix, $x$ and $z$ are column vectors. $M$ is unknown but I know $x$ and $z$. Can I obtain $My$ by $My=Mxx^{-1}y=zx^{-1}y$?
Edited
As pointed out by @Rahul and @Velutluna that I can not do it in this way. If I know $x$ and $z$, how can I compute $My$ without trying to compute $M$? Since in my program large number of iterations evolving this, it will be very time consuming to compute $M$ each time.
Consider the non-zero column vector as an $n \times 1$ matrix. Then of course it has full column rank. Hence the left inverse exists. For $n \ne 1$, it cannot have both left and right inverse. Hence the right inverse appearing in your equation does not exist.