Let $A$ be an $n \times n$ nilpotent real matrix $A$. Define $$e^A = I_n + A + \frac{1}{2!}A^2 + \frac{1}{3!}A^3 + \cdots$$ Choose the correct statement(s) from below: (A) For every real number $t$, $e^{tA}$ is invertible; (B) There exists a basis of $\mathbb{R}^n$ such that $e^A$ is upper-triangular; (C) There exist $B, P \in \mathrm{GL}_n(\mathbb{R})$ such that $B = Pe^AP^{-1}$ and $\operatorname{trace}(B) = 0$; (D) There exists a basis of $\mathbb{R}^n$ such that $A$ is lower-triangular.
Let $A$ be an $n \times n$ nilpotent real matrix $A$. Define
$$e^A = I_n + A + \frac{1}{2!}A^2 + \frac{1}{3!}A^3 + \cdots$$
Choose the correct statement(s) from below:\\
(A) For every real number $t$, $e^{tA}$ is invertible;\\
(B) There exists a basis of $\mathbb{R}^n$ such that $e^A$ is upper-triangular;\\
(C) There exist $B, P \in \mathrm{GL}_n(\mathbb{R})$ such that $B = Pe^AP^{-1}$ and $\operatorname{trace}(B) = 0$;\\
(D) There exists a basis of $\mathbb{R}^n$ such that $A$ is lower-triangular.