Find all solutions in Z16 to the equation $\displaystyle x^2 =9$.
A matrix? Just do the arithmetic!
$\displaystyle 0^2= ?$
$\displaystyle 1^2= ?$
$\displaystyle 2^2= ?$
$\displaystyle 3^2= ?$
$\displaystyle 4^2= ?$
$\displaystyle 5^2= ?$
$\displaystyle 6^2= ?$
$\displaystyle 7^2= ?$
$\displaystyle 8^2= ?$
$\displaystyle 9^2= ?$
$\displaystyle 10^2= ?$
$\displaystyle 11^2= ?$
$\displaystyle 12^2= ?$
$\displaystyle 13^2= ?$
$\displaystyle 14^2= ?$
$\displaystyle 15^2= ?$
All of those are "modulo 16" of course. For example $\displaystyle 10^2= 100= 96+ 4= 6(16)+ 4$ so $\displaystyle 10^2= 4 (mod 16)$.