Let $A=[a_{ij}]$ be an $n \times n$ matrix with $a_{ij}=f_{ij}(x_1,...,x_m)$ where $f_{ij}(x_1,...,x_m)$ is a polynomial in $m$ variables over a finite field $\mathbb{F}_q$.

Let $rank(A)=n$.

Now suppose $(x_i-cx_j)$ divides $determinant(A)$ with $c \in \mathbb{F}_q$ and $rank(A)=n-d$ when $x_i=cx_j$ then does it mean that $(x_i-cx_j)^d$ divides $determinant(A)$ ?

The question can be put more generally where $x_i-cx_j$ can be replaced by an irreducible polynomial $g(x_1,...,x_m)$ where we assume $rank(A)=n-d$ under the assumption $g(x_1,...,x_m)=0$ i.e., we calculate rank of $A$ in quotient space $\mathbb{F}_q[x_1,...x_m]/<g(x_1,...,x_m)>$.

I know that something like above is true for eigenvalues under specific circumstances which is a special case of above. I want to know what happens in general (other than the eigenvalue case).

Thanks