TITLE: Determinant of an $n\times n$ complex matrix as an $2n\times 2n$ real determinant
QUESTION [18 upvotes]: If $A$ is an $n\times n$ complex matrix. Is it possible to write $\vert \det A\vert^2$ as a $2n\times 2n$ matrix with blocks containing the real and imaginary parts of $A$?
I remember seeing such a formula, but can not remember where. Any details, (and possibly references) for such a result would be greatly appreciated.

REPLY [7 votes]: Let $A=B+iC$, where $B$ and $C$ are $n\times n$ real matrices, and let
$$
\widetilde{A}=\left(
\begin{array}{rr}
B & -C \\
C &  B
\end{array}
\right).
$$
Then $\det\widetilde{A}=|\det A|^2$.
Proof.
$$
\begin{align*}
\det\widetilde{A}&=\det\left(
\begin{array}{cc}
B+iC & -C+iB \\
   C &     B 
\end{array}
\right)=
\det\left(
\begin{array}{cc}
B+iC & 0   \\
   C & B-iC 
\end{array}
\right)=
\det
\left(
\begin{array}{ll}
A &  0 \\
C & \overline{A}
\end{array}
\right)\\
&=
(\det A)(\det\overline{A})=(\det A)(\overline{\det A})=|\det A|^2.
\end{align*}
$$