A is an $m \times n$ matrix. If $\operatorname{rank}(A)=n$ and $AX=0$ where $X$ is $n \times k$, then $X = 0$. Below is how I conclude $X = 0$.
Since $\mathsf{rank}(A)=n$ and $A$ is $m \times n$, so we know $A^TA$ is an $n \times n$ invertible matrix. Then $A^TAX$ = $A^T0$, which means $A^TAX = 0$. Then $X = 0$ because $A^TA$ is invertible.
Am I right? If $X$ in $\mathbb{R}^n$, then there is no doubt this statement holds, but $X$ is a matrix, I am not 100% sure.