query: Prove that $\operatorname{trace}(A) = 0$ if and only if $A^2 = 0$.
positive_passages.docid: 150387
positive_passages.text: Questions about matrix rank, trace, and invertibility
negative_passages.docid: 121987
negative_passages.text: Direct proof that nilpotent matrix has zero trace
negative_passages.docid: 31783
negative_passages.text: Is this proof of $ab = 0$ correct?
negative_passages.docid: 11715
negative_passages.text: Prove that if $\langle Tx,x\rangle =0$ for all $x \in X$, then $T = 0$
negative_passages.docid: 52208
negative_passages.text: Show that a matrix $A$ is singular if and only if $0$ is an eigenvalue.
negative_passages.docid: 61274
negative_passages.text: If $AA^T$ is the zero matrix, then $A$ is the zero matrix
negative_passages.docid: 22149
negative_passages.text: How to prove $dim(Ker (A- \lambda I))=dim (Ker(A^t- \lambda I))$?
negative_passages.docid: 25009
negative_passages.text: $\operatorname{tr}(A)=\operatorname{tr}(A^{2})= \ldots = \operatorname{tr}(A^{n})=0$ implies $A$ is nilpotent
negative_passages.docid: 57981
negative_passages.text: Alternative proof of $a\times0= 0$
negative_passages.docid: 154459
negative_passages.text: Constructing a HoTT proof term of 1≠0
negative_passages.docid: 13020
negative_passages.text: Proof for the funky trace derivative : $d (\operatorname{trace} (ABA'C))$?
negative_passages.docid: 75855
negative_passages.text: Show that $\det{(A + sxy^*)}$ is linear in $s$
negative_passages.docid: 55488
negative_passages.text: Proving that $\operatorname{codim}(X,Y)=\dim(\mathcal{O}_{X,\eta})$
negative_passages.docid: 27821
negative_passages.text: A problem on rank of $A$ and $\operatorname{adj}(A)$
negative_passages.docid: 44371
negative_passages.text: Finding trace and determinant of linear operator
negative_passages.docid: 16789
negative_passages.text: Prove that $T^2$ is the zero transformation if and only if $\mathrm{range}(T) \subseteq \ker(T)$.
negative_passages.docid: 88473
negative_passages.text: Proof that the trace of a matrix is the sum of its eigenvalues