$\operatorname{tr}(A^m)=0\implies A$ is Nilpotent
133758
Traces of powers of a matrix $A$ over an algebra are zero implies $A$ nilpotent.
115273
Nilpotent matrix and basis for $F^n$
130268
How to prove that a nilpotent matrix is not invertible?
87494
$\operatorname{rank}(T^3) + \operatorname{rank}(T) \geq 2\operatorname{rank}(T^2)$ for finite-dimensional $V$
7034
Prove that $A+I$ is invertible if $A$ is nilpotent
37867
If $0$ is the only eigenvalue of a linear operator, is the operator nilpotent
106398
If $A$ is invertible show that $\det(A) \neq 0$
41514
Let $trcA=0$.why $A=M+N$ where $M$ and $N$ are nilpotent matrices?
103839
Proving that $T^{2} = T$ for a linear operator on $W$ implies that $V = \operatorname{Null} T \oplus \operatorname{Range} T$
65062
Prove $\text{rank}(A) \geq \frac{(\text{tr}(A))^2}{\text{tr}(A^2)}$ when $A$ is Hermitian
63882
Show that the heisenberg group is nilpotent
68449
$\operatorname{Supp}(M)=V(\operatorname{Ann}M)$ if $M$ is finitely generated
131972
All nilpotent $2 \times 2$ matrices satisfy $A^{2}=0$
52928
If $\mathrm{Tr}(A)=0$ then $T=R^{-1}AR$ has all entries on its main diagonal equal to $0$
47829
Prove that $\operatorname{trace}(A^TA) = 0\ \iff\ A = 0$.
151721
Uhlmann's Theorem: proof of $\text{tr}(A^{\dagger} B) = \langle m | A \otimes B |m\rangle $
48563
If $\Gamma^k_{ij}(p)=0$, then $\nabla_{E_i}E_j (p)=0?$