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

Let $\text{A}$ 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 $\text{A}$ is partitioned into 9 submatrices. The rank of each of these submatices over ${\text{Ff}}_q$, the defining field of $PG(2,q)$, is determined.