# Prove that n is a multiple of $16$.

Let $n$ be a natural number bigger or equal to two and $A,B$ two real matrices of dimension $5$ so that $A^2+B^2=\sqrt[n]{2+\sqrt[n]{2}}AB$ so $\det(AB-BA)>0$. Prove that $n$ is a multiple of $16$.

I don't know how to solve it. I just know that the trace of $AB-BA$ is $0$ and also then, from Cayley Hamilton, I have that $(AB-BA)^2<0$ which is false; from here I have no idea.

• How does $n$ relate to the matrices? – stuart stevenson Mar 8 '18 at 13:36
• I think it's the numerator but the original one it s very ambiguous. It's the only way to relate. – Septimiu Cristian Mar 8 '18 at 13:38
• What about add 2AB and consider det, or trace ? – openspace Mar 8 '18 at 15:08
• Please develop your idea. I don't get it. – Septimiu Cristian Mar 8 '18 at 15:09