Let $A$ be a square invertible matrix which its members are real numbers.
Prove/disprove:
There cannot be a matrix $A$ that satisfies: $A^3+A=0$
I did that:
$$A^3=-A$$ $$A^{-1}A^3=-AA^{-1}$$ $$A^2=-I$$
$$|A|^2=-|I|$$ $$|A|^2\ne-1$$
That is how a prove this, is this right?
because I'm not quite sure about my solution. should I consider a different approach of solving this?
Some help/tips will be highly appreciated :)
Thanks