When $AX=BX \Rightarrow A = B$?

Question

Given the following matrices equation: $AX=BX$ under which assumption we can say that $A=B$?
The obvious one is when $X$ is invertible. Is there any other ?

Answer 1

The matrix $X$ needs to be right invertible.

If it is and $Y$ is a right inverse, so $XY$ is the identity matrix, then $AX=BX$ implies $AXY=BXY$, that is, $A=B$.

On the contrary, if $X$ is not right invertible, it's easy to find examples of matrices $A$ and $B$ with $A\ne B$ and $AX=BX$.

Indeed, if $X$ is $m\times n$ and the rank of $X$ is less than $m$, then there is a column vector $v\ne 0$ such that $X^Tv=0$ (by the rank-nullity theorem and the fact that $X$ and $X^T$ have the same rank). Thus $$ v^TX=0^TX $$

Answer 2

Hint: $AX=BX$ if and only if $AXu=BXu$ for all vectors $u$ (of the right dimension). Now look at the range (column space) of $X$ …