Does rotating a matrix change its determinant? For a $2 \times 2$, it is easy to see the determinant only changes sign.
\begin{align*}
\left(
\begin{array}{cc}
 a & b \\ c & d
\end{array}
\right) \mapsto
\left(
\begin{array}{cc}
 c & a \\ d & b
\end{array}
\right)
\end{align*}
We can see that $\det(A) = -\det(A')$, where $A$ is the original matrix and $A'$ is the rotated matrix. Is this always the case for any $n \times n$ matrix?
Also, this would imply that $\det(A) = \det(A'')$.
Thanks for any advice!

With a $4\times 4$ matrix, rotating preserves the determinant.
In general rotating means transposing (determinant-preserving) followed by turning matrix upside down (multiplies determinant by $(-1)^{\lfloor n/2\rfloor}$).
