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

Hints:
From top-right and bottom-left elements we get:
$$(a+b)d = (a+b)c = -1 \Rightarrow c=-d$$
The top-left and bottom-right elements give us:
$$a^2 + bc = bc + d^2 = 1 \Rightarrow a^2 = d^2 \Rightarrow a = \pm d$$
Can you take it from here?
