We denote by $\mathrm{GL}_2(\mathbb{Z})$ the set of invertible elements of $\mathcal{M}_2(\mathbb{Z})$. The necessary and sufficient condition for $A \in \mathrm{GL}_2(\mathbb{Z})$ to be an $n$-th power is the existence of $\sigma \in \mathbb{Z}$ and $\nu \in \{-1,1\}$ such that $E_{n-1}(\sigma,\nu)$ divides $b$, $c$ and $a-d$, and $\tau = D_n(\sigma,\nu)$, $\delta = \nu^n$. Show that the matrix $A = \begin{pmatrix} 7 & 10 \\ 5 & 7 \end{pmatrix}$ is a cube in $\mathrm{GL}_2(\mathbb{Z})$ and determine a matrix $B \in \mathrm{GL}_2(\mathbb{Z})$ such that $B^3 = A$.
We denote by $\mathrm{GL}_2(\mathbb{Z})$ the set of invertible elements of $\mathcal{M}_2(\mathbb{Z})$. The necessary and sufficient condition for $A \in \mathrm{GL}_2(\mathbb{Z})$ to be an $n$-th power is the existence of $\sigma \in \mathbb{Z}$ and $\nu \in \{-1,1\}$ such that $E_{n-1}(\sigma,\nu)$ divides $b$, $c$ and $a-d$, and $\tau = D_n(\sigma,\nu)$, $\delta = \nu^n$.
Show that the matrix $A = \begin{pmatrix} 7 & 10 \\ 5 & 7 \end{pmatrix}$ is a cube in $\mathrm{GL}_2(\mathbb{Z})$ and determine a matrix $B \in \mathrm{GL}_2(\mathbb{Z})$ such that $B^3 = A$.