TITLE: Is the set $\mathcal{A}$ of all matrices whose trace is $0$ nowhere dense in $\mathbb{M}_n(\mathbb{R}),n \ge 2$?
QUESTION [7 upvotes]: Is the set $\mathcal{A}$ of all matrices whose  trace is $0$  nowhere dense in $\mathbb{M}_n(\mathbb{R}),n \ge 2$ ?

My attempt  : my answer is False
I take $A  = \begin{bmatrix} 1& n \\0&-1 \end{bmatrix}$
I know that set of all invertible matrix is  dense. You know that  my  given matrix $A$ in invertible,  so I  think statement must be false .
Am I right or wrong?
Thank you.

REPLY [3 votes]: Your set is a finite dimensional (and hence closed) proper subspace of $\mathbb{M}_n(\mathbb{R})$ so it is nowhere dense because every proper subspace of a normed space has empty interior.