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

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 '19 at 14:04

score 2 · Accepted Answer · answered Feb 18 '19 at 14:02

2

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.

answered Feb 18 '19 at 14:02

davidlowryduda

91,687

Yeah you are right. – math maniac. Feb 18 '19 at 14:06

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

1 Answers1