Show that $B \in \Bbb M_3 (\Bbb Z).$

Question

Let $A \in \Bbb M_3 (\Bbb Z)$ be such that $A=B^2,$ for some $B \in \Bbb M_3 (\Bbb R).$ Show that $B \in \Bbb M_3 (\Bbb Z).$

Answer 1

Consider the case when $$ B = \left( \begin{matrix} \sqrt 2 & 0 & 0 \\ 0 & \sqrt 2 & 0 \\ 0 & 0 & \sqrt 2 \end{matrix} \right).$$ Then $B^2 = 2I$. It appears your claim is false.