I need help for an exercise from Herstein's Topics in Algebra, 2ed in Sec.6.7 Canonical Form: Rational Canonical Form. Since concepts like trace and determinant are not introduced yet, I wonder how to prove it without using those concepts.
*7. If $A_1,\ldots,A_k$ are in $F_n$ and are such that for some $\lambda_1,\ldots,\lambda_k$ in $K$, an extension of $F$, $\lambda_1A_1+\cdots+\lambda_kA_k$ is invertible in $K_n$, prove that if $F$ has an infinite number of elements we can find $\alpha_1,\ldots,\alpha_k$ in $F$ such that $\alpha_1A_1+\cdots+\alpha_kA_k$ is invertible in $F_n$.