Some $p$-ranks related to a conic in $PG(2,q)$

Let $\A$ be the incidence matrix of lines and points of the classical projective plane $PG(2,q)$ with $q$ odd. With respect to a conic in $PG(2,q)$, the matrix $\A$ is partitioned into 9 submatrices. The rank of each of these submatices over $\Ff_q$, the defining field of $PG(2,q)$, is determined.