# The Number of involutory matrices over $\mathbb{Z_p}$

I want to prove the number of 2-by-2 Involutory Matrices ($$A^2=I$$) over $$\mathbb{Z_p}$$ using quadratic residue and legendre symbol.
I already know that the formula is $$p^2$$ for characteristic of a field is 2 and $$p^2+p+2$$ for $$p$$ being an odd prime as stated here: Involutory matrix $$2 \times 2$$
I have also shown that the number of Involutory Matrices is twice the number of solutions ($$a,b$$) to the congruence : $$a^2+bc \equiv 1$$ (mod $$p$$)
Is there any way to come up with the formula using quadratic residue and legendre symbol? Thanks in advance.