If $a$ is a quadratic residue of odd prime $p$, then is it the case that $a^{-1}$ is also a quadratic residue ?



I notice quadratic residues of odd prime $13$ : 
$$1^2\equiv 12^2\equiv 1\\2^2\equiv 11^2\equiv 4\\3^2\equiv 10^2\equiv 9\\4^2\equiv 9^2\equiv 3\\5^2\equiv 8^2\equiv 12\\6^2\equiv 7^2\equiv 10$$

$4^{-1}=10\\9^{-1}=3\\1^{-1}=1\\12^{-1}=12$ 

Is this true always ? How to go about proving ? thanks! 


Question: Source where you can find if a CVE has a patch or not
Answer: I'm not aware of any good out-of-the-box services, but I usually cross-reference with sites like:


SecurityFocus: It usually references CVE. Patch status filed under "Solution".
PacketStormSecurity: Allows fulltext search (can search for CVE ID). Patch status filed under "Mitigation".
OSVDB: Has an option to search with CVE ID. Patch status filed under "Solution".
IBM XForce: Allows CVE ID search. Patch status filed under "Remediation".