Calculating Eigenvalues from two matrices Let $\alpha$ be the endomorphism given by
$\alpha$:$\mbox{} \left[ \begin{array}{cc}
a & b \\ c & d \end{array} \right]$$\rightarrow$ $\mbox{} \left[ \begin{array}{cc}
 d & -b \\ -c & a \end{array} \right]$
I need to find the eigenvalues and associate eigenspace. Since the determinants are equal, does that mean that the product of the eigenvalues is also the same? And which matrix ought I use to find the eigenvalues? 

maybe relevant
$$
\pmatrix{
1 & 0 \\
0 & 1
}
\to
(+1)
\pmatrix{
1 & 0 \\
0 & 1
}
$$
$$
\pmatrix{
0 & 0 \\
1 & 0
}
\to
(-1)
\pmatrix{
0 & 0 \\
1 & 0
}
$$
$$
\pmatrix{
1 & 0 \\
0 & -1
}
\to
(-1)
\pmatrix{
1 & 0 \\
0 & -1
}
$$
$$
\pmatrix{
0 & 1 \\
0 & 0
}
\to
(-1)
\pmatrix{
0 & 1 \\
0 & 0
}
$$
I would guess that one of the eigenvalues is 1 which corresponds to the 1-dimensional eigenspace
$$
\pmatrix{
x & 0 \\
0 & x
}
$$
and that the other eigenvalue is -1 which corresponds to the 3-dimensional eigenspace
$$
\pmatrix{
y & z \\
w & -y
}
$$
