Let $A=[a\_{ij}]$, where $a\_{ij}=u\_{i}v\_{j}, 1 \leq i \leq n$ and $1\leq j \leq n$ and $u\_i,v\_j$ belong to $R$ satisfies $A^5=16A$. Find trace(A).

I denoted U as a column matrix having values u1,u2,...,un. And V a row matrix having values v1,v2,...,vn. So that A=UV.

But I am not able to proceed further. Evaluating A^5 would be very tedious so I think that I am missing the trick in this question.

Also backtracking from the answer, I feel $A^5=(trace(A))^4A$. Is there any easy way to prove this?