Matrices with all non-zero entries.

I am reading a paper and it uses one of these facts, I would like to know if it has a simple proof:

Let $ F$ be an infinite field and $ n\ge2$ and integer. Then for any non-scalar matrices $ A_1,A_2,…,A_k$ in $ M_{n}(F)$ , there exist some invertible matrix $ Q \in M_{n}(F)$ such that each matrix $ QA_1Q^{-1}, QA_2Q^{-1},…, QA_{k}Q^{-1}$ have all non-zero entries.

I just don’t know where to start at the first place, could have used diagonalizability but not all non-scalar matrices are diagonalizable. Maybe its too simple, please help.

Thanks in advance.