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).

$\begin{pmatrix} a^2 + bc & ab + bd \\  ca + dc & bc + d^2\end{pmatrix}=\begin{pmatrix}1&0\\ 0&1\end{pmatrix}$
You have $b(a+d)=0$ which implies $b=0$ or $a+d=0$
suppose $b=0,a+d\neq0$, you have $a^2=1$ concluding $a=\pm 1$ you can find $d$ and then $c$
suppose $a+d=0, b\neq 0$, you have some other possibility....
You have $c(a+d)=0$ which implies $c=0$ or $a+d=0$ repeating same condition you can find other entities...
