# Show that $B \in \Bbb M_3 (\Bbb Z).$ [closed]

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

## closed as off-topic by user26857, José Carlos Santos, YiFan, uniquesolution, SongMar 12 at 13:04

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• There have been several posts recently on the interaction between $3\times 3$ matrices over $\Bbb Z$ and over $\Bbb R$. What is going on here? – Arthur Feb 18 at 14:04

## 1 Answer

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.

• Yeah you are right. – math maniac. Feb 18 at 14:06