Let $AX=Y$ be a non-homogeneous linear system where $A$ is an $n\times n$ matrix. If this system has a unique solution, can we deduce that $\det A\neq 0$ directly from cramer's rule?
Using the relation $x_i\det A = \det A_i$, if $\det A = 0$ then each $A_i=0$. Then I have no idea of what to do next.
I know that the existence of a non-trivial solution to the homogeneous system $AX=0$ implies that $\det A = 0$. But is the result even true in the non-homogeneous case?