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Let $A$ be complex $2×2$ matrices s.t. $A^2=0$. Which of the following statements are true?

  1. $PAP^{-1}$ is diagonal for some $2×2$ real matrix $P$.
  2. $A$ has $2$ distinct eigenvalues in $\Bbb C$.
  3. $A$ has $1$ eigenvalue in $\Bbb C$ with multiplicity $2$.
  4. $Av=v$ for $v\in \Bbb C^2 ,v≠0$.
  • 0
    where to begin ....please suggest me for 4...2012-12-17

2 Answers 2

1

I will sum the above discussion here:
1. Show that $(PAP^{-1})^2=PA^2P^{-1}$. Since $A^2=0$, what can you conclude about $A$, if indeed there exists such $P$?
2. You said that it is wrong. Can you show why? (Hint: what is the def of an eigenvalue?)
3. Is coorect, as you said. It should follow from the proof you used in 2.
4. If there esists $v\in\mathbb{C}^2$ such that $Av=v$, what is $A^2v$? can that happen?

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    @p.haz: thanks sir.....2012-12-17
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$A$ is a nilpotent matrix, so it has only $0$ as eigenvector with multiplicity $2$.