Nilpotent Matrices

## Primary tabs

# Nilpotent Matrices

Submitted by finles on Thu, 04/10/2003 - 12:41

Forums:

Im just new to linear algebra, and am having trouble proving that a nilpotent matrix must be singular. Can anyone give me a hand in the right direction slash provide the proof?

Cheers

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

Apr 22

new question: Prove that for any sets A, B, and C, An(BUC)=(AnB)U(AnC) by St_Louis

Apr 20

new image: information-theoretic-distributed-measurement-dds.png by rspuzio

new image: information-theoretic-distributed-measurement-4.2 by rspuzio

new image: information-theoretic-distributed-measurement-4.1 by rspuzio

new image: information-theoretic-distributed-measurement-3.2 by rspuzio

new image: information-theoretic-distributed-measurement-3.1 by rspuzio

new image: information-theoretic-distributed-measurement-2.1 by rspuzio

Apr 19

new collection: On the Information-Theoretic Structure of Distributed Measurements by rspuzio

Apr 15

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new question: Prove that for any sets A, B, and C, An(BUC)=(AnB)U(AnC) by St_Louis

Apr 20

new image: information-theoretic-distributed-measurement-dds.png by rspuzio

new image: information-theoretic-distributed-measurement-4.2 by rspuzio

new image: information-theoretic-distributed-measurement-4.1 by rspuzio

new image: information-theoretic-distributed-measurement-3.2 by rspuzio

new image: information-theoretic-distributed-measurement-3.1 by rspuzio

new image: information-theoretic-distributed-measurement-2.1 by rspuzio

Apr 19

new collection: On the Information-Theoretic Structure of Distributed Measurements by rspuzio

Apr 15

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

## Re: Nilpotent Matrices

Let A be a nilpotent matrix. Then by definition for some natural n, A^n = 0.

Let d = |A| (the determinant). |0| = 0 = |A^n| = |A|^n = d^n. Hence, d = 0 and A is singular.

## Re: Nilpotent Matrices

Another way to think about this is to suppose instead that we had an invertible matrix A such that A^n=0. A product of invertible matrices is invertible, so A^n=0 is invertible, but 0 is not invertible.

## Re: Nilpotent Matrices

Yet another approach. Consider N nilpotent (but non-zero) with n the smallest integer (>1) such that N^n = 0. Assume N is non-singular with inverse M, then N^(n-1) = M N^n = M 0 = 0, but n was suppose to be the smallest integer...proof by contradiction.