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Let $k$ be a field and $A_1,\ldots, A_l\in M_n(k)$ be square matrices. Is there any way to determine if $\operatorname{Span}_k\{A_1,\ldots, A_l\}\cap GL_n(k)$ is empty or not?

I am adding more details about the specific cases that I am interested in:

I am interested in the case when $k$ is a finite field. Also, all the $A_i$'s are of the form $$ \left[ \begin{array}{c|c|c|c} C_{11} & C_{12} & \cdots & C_{1m} \\ \hline C_{21} & C_{22} & \cdots & C_{2m} \\ \hline \vdots & & \ddots & \vdots \\ \hline C_{m1} & C_{m2} & \cdots & C_{mm} \end{array} \right]$$ with exactly one block $C_{ij}$ being non-zero.

The dimension for each $C_{ij}$ is given, and each $A_i$ is also given. I am interested in how to determine if $\operatorname{span}\{A_1,\ldots, A_l\}\cap GL_n(k)$ is empty or not. Is there any algorithms with polynomial complexity to determine this?

TH Wang
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  • It is most certainly not empty. $A_{1}\dots,A_{l}$ are square matrices which means there will be certain square matrices which are invertible. Therefore the intersection of all square matrices with invertible square matrices will be non-empty and will consist of all invertible linear square matrices. – JohnColtraneisJC Sep 19 '19 at 22:08
  • @JohnColtraneisJC: what if $n=2$, $l = 1$ and $A_1 = \left(\matrix{1 & 0 \ 0 & 0}\right)$? – Rob Arthan Sep 19 '19 at 22:11
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    By this post, if the span is $n^2 - n + 1$-dimensional with $k \in {\Bbb R, \Bbb C}$, the intersection must be non-empty. – Ben Grossmann Sep 19 '19 at 22:16
  • I meant to say if the span is at least $n^2 - n + 1$-dimensional – Ben Grossmann Sep 19 '19 at 22:29

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