linear algebra – Rank of matrices in a symmetric block matrix?

I Would like to ask the following question:

Given the follwing block matrix $$ X = begin{equation*}
begin{pmatrix}
A_1^TA_1 & A_1^TA_2 & A_1^TA_3 \
A_2^TA_1 & A_2^TA_2 & A_2^TA_3 \
A_3^TA_1 & A_3^TA_2 & A_3^TA_3 \
end{pmatrix}
end{equation*}.$$

where $A_i in mathbb{C}^{Mtimes r}$, $text{ rank($A_i$) = }r, forall i = 1,2,3, text{ and } M>r$.

Now, one can write the above matrix, using column operations, as:

$$ X = begin{equation*}
begin{pmatrix}
I & B & C \
0 & I & D \
0 & 0 & I \
end{pmatrix}
end{equation*}.$$

My question is: what could we say about the rank of the matrices $B, C, D$?