# Transformations that leave linear program unchanged

A typical linear program is written as L0: $$min_{x \geq 0; A^T x \leq b}$$ $$c^Tx$$. Here, $$x \in \mathcal{R}^n$$, $$c \in \mathcal{R}^n$$, $$A \in \mathcal{R}^{m \times n}$$, and $$b \in \mathcal{R}^m$$.

Now consider a matrix $$M \in \mathcal{R}^{m \times m}$$ which is multiplied on both sides of the inequality constraints such that the problem becomes L1: $$min_{x \geq 0; MA^T x \leq Mb}$$ $$c^Tx$$.

What should be the constraints on M such that L0 and L1 have the same

1. feasible region
2. and/or the same solutions

The answer is somewhat clear in the case of equality - M should be an invertible matrix (I might be wrong here too).

What mathematical concepts should I use to reason about the above problem? Any help is highly appreciated.