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Let $A$ be a real symmetric matrix having $0$ and $1$ as its only eigenvalue. Let the dimension of the null space of $A-I=m$. Pick out the true option(s):

  1. Characteristic polynomial of $A$ is $(\lambda-1)^m\lambda^{m-n}$

  2. $A^k=A^{k+1} \forall k$

  3. Rank of $A$ is $m$

My try:

  1. Since a real symmetric matrix is diagonalisable, its minimal polynomial must have distinct roots. Since 0 and 1 are the only roots possible, $x(x-1)$ is the only possibility. Thus it follows that $A^k=A^{k+1}$.

  2. Rank of a matrix is the number of non-zero eigenvalues which is $m$ in this case. So the rank of $A$ is $m$.

For 1 I am not quite sure. Please find my errors if any and suggest required edits.

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    $\begingroup$ I see that you already posted a large number of questions of the form "Pick out..." (math.stackexchange.com/questions/1105507/…, math.stackexchange.com/questions/1099050/…, math.stackexchange.com/questions/1055192/pick-out-compact-sets, math.stackexchange.com/questions/1054003/…, ...). This type of question is not very search-friendly, as such a question contains many questions inside. [...] $\endgroup$ – zarathustra Jan 17 '15 at 8:56
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    $\begingroup$ [...] Moreover, when you have been answered, you should accept an answer (I can't imagine that for all these questions, you never had an appropriate answer). Finally, by now, you should know what are the "required edits" and not expect the community to do all the work for you. $\endgroup$ – zarathustra Jan 17 '15 at 8:56
  • $\begingroup$ @zarathustra, thank you for finding the old questions so i won't bother answering. $\endgroup$ – abel Jan 17 '15 at 9:50
  • $\begingroup$ @zarathustra why are you saying the question has been answered? Where have u seen them $\endgroup$ – Learnmore Jan 17 '15 at 13:27
  • $\begingroup$ Among the four I listed, two have answers. There are other questions of yours that have answers, but where you don't have accepted any of them. $\endgroup$ – zarathustra Jan 17 '15 at 13:33

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