Linear combination of matrices over finite and infinite fields

Question

Let $F\subset K$ be the fields. Let $A_1,\ldots, A_m$ be the $n\times n$ matrices over the field $F$, and $c_1,\ldots,c_m\in K$ such that $c_1A_1+\cdots+c_mA_m$ is invertible. How to prove that for infinite $F$ there exists $f_1,\ldots,f_m\in F$ such that $f_1A_1+\cdots+f_mA_m$ is invertible and for finite $F$ it does not hold?

Answer 1

$\det(X_1 A_1 + \cdots + X_m A_m)$ is a polynomial, homogeneous of degree $n$, in $F[X_1,\ldots , X_m]$.

If $F$ is infinite, then a polynomial which is identically zero on $F$ must be the zero polynomial, hence zero on $K$.

If $F$ is finite of cardinality $q$, then $XY^q - X^q Y$ is identically zero on $F$. So we can look for a counterexample with $m=2$, $n=q+1$. (Hint: there is such a counterexample where both $A_1$ and $A_2$ are diagonal.)