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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

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I realize this is an answer to a problem I had a while back, but it might be of use to other people who come across a similar question like this one. But, after looking this up more thoroughly, I have concluded that Cramer's Rule works the same with real and complex numbers. Here is the reference.

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 $x=i,\ y=i-1.$

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  • $\begingroup$ I think the the value of y = 1 - i $\endgroup$ – James caps Aug 22 '18 at 14:34

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