Consider an $n \times n$ matrix $\bf A$ over a field. Let $\bf A$ is constructed by the product of $n \times n$ matrices $B_i$, for $1\leq i \leq m$ which means $$ {\bf A}=\prod_{i=1}^m\, {\bf B}_i\, . $$ From Linear Algebra we know that the relation between determinant of $\bf A$ and $\bf B_i$'s are as follows: $$ {\det(\bf A)}=\prod_{i=1}^m\, {\det(\bf B}_i)\, . $$ Suppose that an $k \times k$ sub-matrix of $\bf A$ for $1\leq k \leq n$ is denoted with ${\bf A}^{(k)}$.
My question:
Is there a condition that can be used to check ${\bf A}^{(k)}$ is singular or non-singular matrix based on the determinants of $\bf B_i$'s or sub-matrices of $\bf B_i$'s.
Thanks for your assistance in advance.