Invertible matrices property

Question

I'm wondering about this property :

$\forall A \ \in \ \mathbb{M}_n(\mathbb{R})$ if there is $k \ \in \ \mathbb{N}^{*}$ such as $A^k=I_n$ then $A$ is invertible.

I think this assertion is true I do not find counter-examples but I do not see how to proove it. Would you have an idea ?

Thank you

This is obvious since $AA^{k-1}= A^{k-1}A=I_n$ , so that the inverse is exactly $A^{k-1}$. — Crostul, Mar 10 '15 at 19:53
This claim can easily be proven by induction, it's actually a special case of a similar result for groups. — Mathemagician1234, Mar 10 '15 at 19:55

score 4 · Answer 1 · answered Mar 10 '15 at 19:53

4

For matrices with your property, $AA^{k-1}=A^{k-1}A=I$

answered Mar 10 '15 at 19:53

paw88789

1 Answers1