Finding the determinant of $2A+A^{-1}-I$ given the eigenvalues of $A$ 
Let $A$ be a $2\times 2$ matrix whose eigenvalues are $1$ and $-1$. Find the determinant of $S=2A+A^{-1}-I$.

Here I don't know how to find $A$ if eigenvectors are not given. If eigenvectors are given, then I can find using $A=PDP^{-1}$. Please solve for $A$; the rest I can do.

\begin{pmatrix}
0 &-1 \\ 
 -1&0 
\end{pmatrix}
the Characteristic polynomial is $(x-1)(x+1)$. 
