This is my thesis problem: Determine the number of $2\times2$ skew orthogonal matrix in $\mathbb{Z}_p$ for $p>2$.
$A$ is skew orthogonal if $A^tA=-I$, where $A^t$ denotes the transpose and $I$ is the identity matrix.
I need to obtain the characteristic polynomial of a skew orthogonal matrix. Here's my findings:
$2\times2$ general form of characteristic polynomial: $x^2-\operatorname{tr}(A)x+\det(A)$.
$\det(A^tA)=\det(-I)$
$(\det A)^2=1$
$\det A= 1,-1$
But I don't know the trace of $A$ such that $A^tA=-I$ in $\mathbb{Z}_p$ for $p>2$.
Hint: From Cayley-Hamilton theorem you have
$A^2-\text{tr}(A)A+\det(A)I=0$.
Multiply both sides by $A^T$ to obtain
$(A^TA)A-\text{tr}(A)(A^TA)+\det(A)A^T= -A+\text{tr}(A)I+\det(A)A^T=0 $.
Hence $\text{tr}(A)I=A-\det(A)A^T$.
What conclusion can be made from this equation?