query: $C=AB-BA$. If $CA=AC$, then $C$ is not invertible.
positive_passages.docid: 133482
positive_passages.text: $AB-BA$ is a nilpotent matrix if it commutes with $A$
negative_passages.docid: 8787
negative_passages.text: In a ring $(A,+, \cdot)$ if $aba = a$ then $bab = b$ and all non zero elements in $A$ are invertible.
negative_passages.docid: 110944
negative_passages.text: Proving that $AB-BA=cI$ for nontrivial $c \in \mathbb{C}$
negative_passages.docid: 157087
negative_passages.text: Can anyone explain why $a^{b^c} = a^{(b^c)} \neq (a^b)^c = a^{(bc)}$
negative_passages.docid: 586
negative_passages.text: What exactly are we doing when we set $c=1$?
negative_passages.docid: 124868
negative_passages.text: Prove that $\frac{bc}{b+c}+\frac{ca}{c+a}+\frac{ab}{a+b}<\frac{a+b+c}{2}$
negative_passages.docid: 136367
negative_passages.text: Prove $ ∀a, b, c ∈ Z,$ if $ab + ac ≡ 3$ (mod 6) then b $\not\equiv$ c (mod 6).
negative_passages.docid: 7622
negative_passages.text: $AB-BA=I$ having no solutions
negative_passages.docid: 158502
negative_passages.text: Showing $\sin(\bar{z})$ is not analytic at any point of $\mathbb{C}$
negative_passages.docid: 32253
negative_passages.text: How to prove $b=c$ if $ab=ac$ (cancellation law in groups)?
negative_passages.docid: 81868
negative_passages.text: How to prove $\operatorname{Tr}(AB) = \operatorname{Tr}(BA)$?
negative_passages.docid: 23869
negative_passages.text: Prove the implication "if $ac>bc$ and $c>0$, then $a>b$" using the contrapositive
negative_passages.docid: 125668
negative_passages.text: A proof that BS(1,2) is not polycyclic
negative_passages.docid: 88349
negative_passages.text: Are the eigenvalues of $AB$ equal to the eigenvalues of $BA$? (Citation needed!)
negative_passages.docid: 150282
negative_passages.text: If $A$ is independent of $B$ and $C$ then why is it not necessarally independent of $B\cap C$?
negative_passages.docid: 155750
negative_passages.text: Why does $\mathbb{C}$ have transcendence degree $\mathfrak{c}$ over $\mathbb{Q}$?
negative_passages.docid: 40663
negative_passages.text: $a_1 a_1', a_2 a_2', ..., a_n a_n'$ is not a reduced modulo system $\mod p$.