Solve $X^2=A$, where X is a 2 by 2 matrix and A is a known matrix

Question

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

Answer 1

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?

Answer 2

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.