Let $\mathcal{M}_{n}$ denote the vector space of $n \times n$ matrices over a field $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$. What is the $\operatorname{rank} r(A)$ of a matrix $A \in \mathcal{M}_{n}$ ?

Show, stating accurately any preliminary results that you require, that $r(A)=n$ if and only if $A$ is non-singular, i.e. $\operatorname{det} A \neq 0$.

Does $\mathcal{M}_{n}$ have a basis consisting of non-singular matrices? Justify your answer.

Suppose that an $n \times n$ matrix $A$ is non-singular and every entry of $A$ is either 0 or 1. Let $c_{n}$ be the largest possible number of 1 's in such an $A$. Show that $c_{n} \leqslant n^{2}-n+1$. Is this bound attained? Justify your answer.

[Standard properties of the adjugate matrix can be assumed, if accurately stated.]