query: Proof that if $\mathrm{tr}\,A^k=0$ for all $k=1,\ldots, n$, then $A^n = 0$
positive_passages.docid: 25009
positive_passages.text: $\operatorname{tr}(A)=\operatorname{tr}(A^{2})= \ldots = \operatorname{tr}(A^{n})=0$ implies $A$ is nilpotent
negative_passages.docid: 29462
negative_passages.text: Suggestions on how to prove the following equality. $a^{m+n}=a^m a^n$
negative_passages.docid: 7843
negative_passages.text: Show that order of $a^k$
negative_passages.docid: 41514
negative_passages.text: Let $trcA=0$.why $A=M+N$ where $M$ and $N$ are nilpotent matrices?
negative_passages.docid: 147014
negative_passages.text: Matrix algebra: If $A^2=0$, Proof rank(A) $\le \frac{n}{2}$
negative_passages.docid: 115782
negative_passages.text: Proof $\mathbb{A}^n$ is irreducible, without Nullstellensatz
negative_passages.docid: 47670
negative_passages.text: Algebraic proof of $\sum_{i=0}^k{{n \choose i}{m \choose {k-i}}}= {{m+n}\choose k}$
negative_passages.docid: 154459
negative_passages.text: Constructing a HoTT proof term of 1≠0
negative_passages.docid: 34714
negative_passages.text: What is the relationship between $\ker(A), \ker(A^2), \ldots, \ker(A^n)$?
negative_passages.docid: 106584
negative_passages.text: If $A^{2n}=I$, then $A^n=\pm I$ where $A\in M_n(\mathbb{R})$
negative_passages.docid: 21781
negative_passages.text: Prove for any $a>1$ there exist a natural number, $N$, such that $a^n>n$ for all $n \ge N$
negative_passages.docid: 11895
negative_passages.text: If $AB=0$ prove that $\mathrm{rank}(A)+\mathrm{rank}(B)\leq n$
negative_passages.docid: 55461
negative_passages.text: How to prove that if $a$ belongs to $\mathbb R$, such that $0\leq a \leq\epsilon$, then $a = 0$
negative_passages.docid: 67645
negative_passages.text: Prove that $\displaystyle{\lim_{n \to \infty}a_n ^{1/k}}= a^{1/k}$ if $a_n \ge 0$ for all $n$ and $\displaystyle{\lim_{n \to \infty}a_n}= a$
negative_passages.docid: 57981
negative_passages.text: Alternative proof of $a\times0= 0$
negative_passages.docid: 35897
negative_passages.text: Proof of: $AB=0 \Rightarrow Rank(A)+Rank(B) \leq n$
negative_passages.docid: 15259
negative_passages.text: Proving $\prod \limits_{k=0}^{n}(1-a_k) \geq1- \sum\limits_{k=0}^{n}a_k$