Articles of nilpotence

How to find a nonzero $2 \times 2$ matrix whose square is zero?

How do I find a nonzero $2 \times 2$ matrix $A$ such that $A^2$ has all zeros entries? Very confused with this. Could use all the help I can get. Thank you

Jointly nilpotent matrices

We say that a matrix $J \in \mathbb{R}^{n \times n}$ is nilpotent if $J^n = 0$. This is equivalent to the statement that $\forall x \in \mathbb{R}^{n} \quad \exists k \in \mathbb{Z}^+$ such that $J^kx = 0$. What I would like to do is to extend this notion to pairs of matrices in the following […]

Lifting idempotents modulo a nilpotent ideal

The problem is this: Suppose $I \subseteq R$ is a nilpotent ideal and there is $r \in R$ with $r \equiv r^2 \pmod I$. Show $r \equiv e \pmod I$ for some $e \in R$ idempotent. I have spent a few hours rolling around in abstracta with no destination. I believe that if I could […]

degree of nilpotent matrix

I need to prove or disprove that A nilpotent matrix’s degree is less than or equal to its dimension. I tried to make a counterexample but I found nothing and I think that this claim is true but I don’t know the start point!

The kernel of $R \to A \otimes_R B$ is nil

Let $R \to A$ and $R \to B$ be two homomorphisms of commutative rings whose kernels are nil (i.e. consist only of nilpotent elements). Then the kernel of $R \to A \otimes_R B$ is also nil. See SE/916173 for Zhen Lin’s proof of this fact, which uses algebraic geometry, notably Chevalley’s Theorem. This is used […]

Decompose this matrix as a sum of unit and nilpotent matrix.

Show that the matrix $A=\begin{bmatrix} 1 & 0 \\ 2 & 1 \\ \end{bmatrix}$ can be decomposed as a sum of a unit and nilpotent matrix. Hence evaluate the matrix $A^{2007}$. I read about nilpotent matrices the other day. It said “A square matrix such that is the zero matrix for some positive integer matrix […]

Let $A$ and $B$ be square matrices of the same size such that $AB = BA$ and $A$ is nilpotent. Show that $AB$ is nilpotent.

A square matrix $A$ is called nilpotent is $A^k=0$ for some positive integer $k$. Let $A$ and $B$ be square matrices of the same size such that $AB = BA$ and $A$ is nilpotent. Show that $AB$ is nilpotent. Following that is $AB$ not equal to $BA$ in part (b), must $AB$ be nilpotent? I […]

An exercise about nilpotent quotient groups

Let $G$ be a finite group and $N$ be a normal subgroup of $G$ such that $G/N$ is nilpotent. a) Prove that there exists a nilpotent subgroup $K$ of $G$ such that $G=NK.$ b) Suppose that $N$ is abelian and $Z\left(G\right) =\lbrace e \rbrace.$ Show that if $K$ is nilpotent and $G=NK$ then $K=N_G\left(K\right)$ and […]

$1+a$ and $1-a$ in a ring are invertible if $a$ is nilpotent

This question already has an answer here: Units and Nilpotents 3 answers