Supposing determinants of $A_i=\begin{bmatrix}a_i&b_i\\c_i&d_i\end{bmatrix}$ from $i=1$ to $4$ and $A=\begin{bmatrix}a&b\\c&d\end{bmatrix}$ are $0$ over $\Bbb F_p$ for large $p$. Then can rank of $\begin{bmatrix}aA_1&bA_2\\cA_3&dA_4\end{bmatrix}$ over $\Bbb F_p$ for large $p$ be $4$?
Please take $$a_i\neq b_i\neq c_i\neq d_i$$$$a\neq b\neq c\neq d$$ which is my situation and no entries are $0$.
What can be its maximum rank?
Denote $E_{ij}$ the matrix that has $(i, j)$ entry $1$ and all other entries zero. Setting $A_1 = E_{11}, A_2 = E_{21}, A_3 = E_{22}, A_4 = E_{12}$ and $A = \pmatrix{1&1\\1&1}$ yields the (permutation) matrix $$ \left(\begin{array}{cc|cc} 1 \\ & & 1 \\ \hline & & & 1 \\ & 1 \end{array} \right) . $$