Linear Algebra question: Suppose G is a 2x2 matrix with real entries, such that det(G)=+/- 1 and G^2 = - I (I = id 2x2 id matrix)

[Steven G]

Suppose G is a 2x2 matrix with real entries, such that det(G)=+/- 1 and G^2 = - I (I = id 2x2 id matrix)
What can G be? I came with matrices of the following form. Zeros on the main diagonal and b and -1/b on the off diagonal where b is any non-zero real number.

I was wondering if you can show how you would have this problem as it took me much too long to get my answer and I might have even missed some cases.
Thanks for your time!

 

[blamocur]

In addition to your result I got another family of matrices:
$$G = \begin{pmatrix} x & y \\ -\frac{1+x^2}{y} & -x\end{pmatrix}$$where $x$ is an arbitrary real number and $y\neq 0$. Note that it contains your result when $x=0$.

 

[blamocur]

A side note: if $\det(G)=-1$ then $G$ cannot be real, and all solutions have forms
$$G = \begin{pmatrix}\pm i & y \\ 0 & \pm i\end{pmatrix} \;\;\;\text{or}\;\;\;\; G = \begin{pmatrix}\pm i & 0 \\ y & \pm i\end{pmatrix}$$