Name for matrix operation that inverses off-diagonal elements Is there any operation that sends matrix
$$
S=\left(\begin{array}{cc} a&b\\
c& d
\end{array}\right) \to S' 
=\left(\begin{array}{cc} \:\:\:a&-b\\
-c& \:\:\:d
\end{array}\right)
\quad a,b,c,d \in \mathbb{R}
$$
by setting off-diagonal elements of 2x2 matrix to inverse?
My question is: is there a name (or notation) for this kind of matrix operation?
If I would assume that $b,c \in Im$ (imaginary) then the operation obviously can be defined as conjugate $S'=(S^*)^T$. But all values are real in the given case.
Edit: $S \in SL(2, \mathbb{R}) $

The two matrices $S$ and $S'$ are conjugate, or similar since they have the same characteristic polynomial
$$
t^2-(a+d)t+(ad-bc),
$$
provided that they aren't a scalar multiple of the identity, i.e., provided that $(b,c)\neq (0,0)$.
