Solve $X^2=A$, where X is a 2 by 2 matrix and A is a known matrix A = $
        \begin{pmatrix}
        1 & 1 \\
        -1 & 1 \\
        \end{pmatrix}
$
. I wrote X = $
        \begin{pmatrix}
        a & b \\
        c & d \\
        \end{pmatrix}
$. So $X^2$=$
        \begin{pmatrix}
        a^2+bc & ab+bd \\
        ac+bc & bc+d^2 \\
        \end{pmatrix}
$ = $
        \begin{pmatrix}
        1 & 1 \\
        -1 & 1 \\
        \end{pmatrix}
$
Unfortunatelly I don't know how to continue from here and maybe someone can help be find the matrix X

You get:
$a^2 +bc$ = 1
$ac + bc$ = -1
$ab + bd$ = 1
$bc + d^2$ = 1
Hence use substitution to find a, b, c and d. 
