If $AX=B$, does $\det(AX)=\det(B)$?
The equation $AX=B$ says that $AX$ and $B$ are two expressions for the same matrix, so of course $f(AX)=f(B)$ for any operation $f$. It's the same thing on both sides.
As pointed out in the comments, the linear algebra fact being used is that the determinant is multiplicative, so that $\det(AX)=\det(A)\det(X)$.