Solve the matrix equation $X ^ 3 = A$, with $X \in M_2(\mathbb{R})$ and given $A$.
The matrix $A$ has $2$ eigenvalues: $1$ and $3$. The vectors $(1,-1)$ and $(1,1)$ are eigenvectors that correspond to the eigenvalues $1$ and $3$ respectively. Therefore, if$$P=\begin{bmatrix}1&1\\-1&1\end{bmatrix}$$(the colums of $P$ are the eigenvectors), then$$P^{-1}AP=\begin{bmatrix}1&0\\0&3\end{bmatrix}.$$So, take\begin{align}X&=P\begin{bmatrix}1&0\\0&\sqrt[3]3\end{bmatrix}P^{-1}\\&=\frac12\begin{bmatrix}\sqrt[3]3+1&\sqrt[3]3-1\\\sqrt[3]3-1&\sqrt[3]3+1\end{bmatrix}.\end{align}