Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

$S$ is a non-zero $3$ by $3$ matrix. Is the statement "$S^4 = 0$ but $S^3 \neq 0$" necessarily false?

share|improve this question
4  
Hint: What could be the characteristic polynomial of $S$? –  Did Oct 30 '12 at 13:27

2 Answers 2

up vote 1 down vote accepted

Since $S^4=0$, $S$ is a nilpotent. The degree of an n × n nilpotent matrix is always less than or equal to n, so $S^3=0$.

share|improve this answer
    
Thanks! Though I have taken introductory linear algebra, we never did nilpotent matrices. So, I didn't know this property! :/ –  Legendre Oct 30 '12 at 13:58
    
Oh nevermind :) I missed that you also specified it was 3x3! –  rschwieb Oct 30 '12 at 14:04

Since you know that $x^4$ is an annihilating polynomial of $S$, that means the minimal polynomial divides $x^4$. Since the minimal polynomial is always of smaller (or equal) degree to the characteristic polynomial, we know that it must be one of $$\left\{x,\ x^2,\ x^3\right\}$$ In particular this shows that the characteristic polynomial must be $p(x)=x^3$ where Cayley-Hamilton shows that $S^3 = 0$. You cannot correctly conclude that $S^2 = 0$ however, for example $$S=\begin{pmatrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0\end{pmatrix}$$ Satisfies $S^3 = 0$ but $S^2 \neq 0$.

share|improve this answer
    
My mistake was answering the title question correctly, but failing to read the additional hypotheses mentioned in the OP :/ –  rschwieb Oct 30 '12 at 14:07
    
@rschwieb We're all guilty of that sometimes! :) –  EuYu Oct 30 '12 at 14:10

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.