Solve $x^2 = I_2$ where x is a 2 by 2 matrix I tried a basic approach and wrote x as a matrix of four unknown elements $\begin{pmatrix} a && b \\ c && d \end{pmatrix}$ and squared it when I obtained $\begin{pmatrix} a^2 + bc && ab + bd \\  ca + dc && cd + d^2\end{pmatrix}$ and by making it equal with $I_2$ I got the following system
$a^2 + bc = 1$
$ab +bd = 0$ 
$ca + dc = 0$
$cd + d^2 = 1$
I don't know how to proceed. (Also, if anyone knows of a better or simpler way of solving this matrix equation I'd be more than happy to know).

The second and third equalities are 
$$b(a+d)=0$$
$$c(a+d)=0$$
Split the problem in two cases:
Case 1: $a+d \neq 0$. Then $b=0, c=0$, and from the first and last equation you get $a,d$.
Case 2: $a+d=0$. Then $d=-a$ and $bc=1-a^2$. Show that any matrix satisfying this works.
