Putnam 2010 B6

Let $A$ be an $n \times n$ matrix of real numbers for some $n \geq 1$. For each positive integer $k$, let $A^{[k]}$ be the matrix obtained by raising each entry to the $k$th power. Show that if $A^k = A^{[k]}$ for $k=1,2,\dots,n+1$, then $A^k = A^{[k]}$ for all $k \geq 1$.