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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 $v\in \Bbb C^2 ,v≠0$.
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    not getting any A s.t. A^2=0.2012-12-17
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    where to begin ....please suggest me...........2012-12-17
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    Take $A=\begin{pmatrix}a&a\\-a&-a\end{pmatrix}$ for any $a\in\mathbb{C}$, for exmaple.2012-12-17
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    2 is wrong ........ 3 is correct..........2012-12-17
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    That's right. Can you prove it?2012-12-17
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    what about 1 and 4..............2012-12-17
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    Check that $(PAP^{-1})^2=PA^2P^{-1}$. What can you conclude about 1?2012-12-17
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    I can't find 4 but1 is looking right2012-12-17
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    where to begin ....please suggest me for 4...2012-12-17

2 Answers 2