Properties of upper triangular matrix

Question

let $A$ be the set of all invertible upper triangular matrics in $\mathbb{M}_n (\mathbb{R})$ where $n \ge 2$

then $A $ is

choose the correct option

$1.$ dense

$2.$Nowheredense

$3.$open

$4.$ closed

My attempt :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 and upper triangular, so $ A$ must be open and dense

Therefore the correct option is option $1)$ and option $3)$

is its true ?

Any hints/solution will be appreciated

thanks u

Answer 1

Since $\overline A$ is the set of all upper triangular matrices (invertible or not), which is strictly larger than $A$, $A$ is neither closed nor dense. Also, $A$ is not open because $\operatorname{Id}\in A$ and every neighborhood of $\operatorname{Id}$ (and, in fact, of any element of $A$) contains non-triangular matrices. In fact, $A$ is nowhere dense (that is, the second option is the correct one), because the interior of $\overline A$ is empty (every neighborhood of any element of $\overline A$ has non-triangular matrices).