Nilpotent binary matrices over finite fields

Question

I am studying on nilpotent matrices over finite fields. By definition a square matrix $A$ is $p$-nilpotent if a power of $A$ modulo $p$ is the zero matrix. For example. Let $J_2$ be the $2\times 2$ matrix of ones. Then $J_2$ is 2-nilpotent. Now, if we consider a $p\times p$ matrix $A$ in which entries are from the set $\{0,1\}$ ($A$ is usually called binary or $(0,1)$-matrix) and $A$ is $p$-nilpotent so that the entries all the main diagonal of $A$ are all ones. Does any one has any idea about the form of matrix $A$? I believe that in this case $A$ should be equal to the all ones matrix.