Find all possible real 2x2 matrices A I am trying to find all possible real $2×2$ matrices $A$ such that $A^{2019} = \begin{bmatrix}
     0 &  1 \\
    -1 & 0
\end{bmatrix}$
I am really stuck with this one, could someone provide some insight to help me solve it? Thanks!
 A: Hint Let $A= \begin{bmatrix} x & y \\
z &t \end{bmatrix}$. Then
$$\begin{bmatrix}
     0 &  1 \\
    -1 & 0
\end{bmatrix}\begin{bmatrix} x & y \\
z &t \end{bmatrix}=A^{2019}A=AA^{2019}=\begin{bmatrix} x & y \\
z &t \end{bmatrix}\begin{bmatrix}
     0 &  1 \\
    -1 & 0
\end{bmatrix}$$
This gives,
$$\begin{bmatrix}
     z &  t \\
    -x & -y
\end{bmatrix}=\begin{bmatrix}
     -y &  x \\
    -t & z
\end{bmatrix}$$
This implies $z=-y$ and $t=x$, and hence $A$ must be of the form $$\begin{bmatrix} x & y \\
-y &x \end{bmatrix}$$
Now, identify $A$ with a complex number.
A: We may identify $$ \begin {bmatrix}a&b\\-b&a\end{bmatrix}$$ with the complex number $$a+bi$$
Thus the matrix $$ \begin {bmatrix}0&1\\-1&0\end{bmatrix}$$ will be identified with $i$
The problem is finding all $2019$ complex numbers $z$ such  that $$z^{2019}=i=e^{i \pi /2 }$$ 
These are $$\cos (\theta_k ) + i \sin (\theta_k)$$ for $$\theta_k = \frac {\pi/2 +2k\pi }{2019}$$ for $k=0,1,2,...,2018$
The matrices then are $$ \begin {bmatrix}\cos \theta_k&\sin \theta_k\\-\sin \theta_k&\cos \theta_k\end{bmatrix}$$ 
