# Cramer's Rule with complex system of equations

Given a 2x2 system of complex equations with one unknown, $z$, written as a 2x2 matrix, $A$, would the system have infinitely many solutions iff $\det(A_x)=\det(A_y)=\det(A)=0$? Or is there more to it than just finding the conditions under which the system is indeterminate? I guess what I'm asking is it the same case as when dealing with real valued systems of equations? Any help is appreciated along with some references if possible. Thanks

Example. We can use Cramer's rule to solve this system of equations: $$\begin{cases}ix+2y=1-2i\\4x-iy=-1+3i\end{cases}$$ Using Cramer's Rule we get $$x=\frac{\begin{vmatrix}1-2i&2\\-1+3i&-i\end{vmatrix}}{\begin{vmatrix}i&2\\4&-i\end{vmatrix}}=\frac{-i(1-2i)-2(-1+3i)}{i(-i)-2(4)}=\frac{-7i}{-7}=i\\y=\frac{\begin{vmatrix}i&1-2i\\4&-1+3i\end{vmatrix}}{\begin{vmatrix}i&2\\4&-i\end{vmatrix}}=\frac{i(-1+3i)+4(1-2i)}{i(-i)-2(4)}=\frac{-7+7i}{-7}=i-1$$ Therefore, the solution is $x=i,\ y=i-1.$