Is set $\{A \in \mathbb{R}^{2 \times 2} : \text{tr}(A) = 0\}$ is bounded and closed?

Question

$$\{A \in \mathbb{R}^{2 \times 2} : \text{tr}(A) = 0\}$$ Inner product is defined to be <$A, B$>$ = \text{tr}(A^TB) $ I was asked if the set $$D = s \subset \text{Mat}_{2×2}$$ is closed and bounded. I showed that $D^c$ is open hence $D$ closed (is it correct?). how do I show It's (not) bounded? Thank you

Answer 1

Hint What is the norm of $$A = \begin{bmatrix} 0 & n \\ n& 0\end{bmatrix}$$?