# 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.