<s>[INST] Complex numbers inside determinant Let $ \begin{vmatrix}6\iota & -3\iota & 1\\ 4 & 3\iota & -1\\ 20 & 3 & \iota \\ \end{vmatrix}= x +\iota y$, then what are the values of $x$ and $y$?
 [/INST] If you did not notice the linear dependency in the last two columns, then the "standard way" should have given you:
$$\begin{align}\begin{vmatrix}6\iota & -3\iota & 1\\ 4 & 3\iota & -1\\ 20 & 3 & \iota \\ \end{vmatrix}
=&~ 6\iota\begin{vmatrix}3\iota&-1\\3&\iota\end{vmatrix}
-3\iota\begin{vmatrix}-1&4\\\iota&20\end{vmatrix}
+1\begin{vmatrix}4&3\iota\\20 & 3\end{vmatrix} 
\\[1ex] =&~ 6\iota(3\iota^2+3)-3\iota(-20-4\iota)+(12-60\iota)
\\[1ex] =&~ 18\iota^3+12\iota^2+18\iota+12\\[1ex] =&~ 0\end{align}$$
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