Rotating Matrix by $180$ degrees through another matrix To rotate a $2\times2$ matrix by $180$ degrees around the center point, I have the following formula:
$PAP$ = Rotated Matrix, where
$$P =\begin{bmatrix}
0 & 1\\ 
1 & 0
\end{bmatrix}$$
$$A= \begin{bmatrix}
a & b\\ 
c & d
\end{bmatrix}$$
And the resulting matrix will equal
\begin{bmatrix}
d & c\\ 
b & a
\end{bmatrix}
I need to have this in the form of:
$AP$ = Rotated matrix.
How would I get it to this form?

There are matrices $A$ for which no such $P$ exists. For example, if $$A=\left[\begin{array}{cc}0 & 0\\c & d\end{array}\right]$$ for some $c,d$ not both $0$, then for any $$P=\left[\begin{array}{cc}\alpha & \beta\\\gamma & \delta\end{array}\right]$$ we have $$AP=\left[\begin{array}{cc}0 & 0\\c\alpha+d\gamma & c\beta+d\delta\end{array}\right]\neq\left[\begin{array}{cc}d & c\\0 & 0\end{array}\right].$$
Our problem, then, is worse than the impossible task of trying to find a single matrix $P$ that rotates all matrices $A$ in this way--sometimes, we can't even rotate a given matrix like this.
