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$a$ in that case $T$ is not invertible hence $T$ is not injective and surjective. hence $(a)$ is true statement. am I right?

$b$ need not be.

$c$ is true as a three dimensional eigen space is invariant hence if I restrict characteristic polynomial on that space it is three degree must have a real root.is my intuition right? please help.

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All sound good. –  user1551 Dec 18 '12 at 8:10
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up vote 1 down vote accepted

Looks fine but I'm afraid your answer to (b) won't be so well-accepted, in spite of being correct: in mathematics, in order to show a general statement in false, you must give a counterexample.

I'd say to look at something like

$$T=\begin{pmatrix}2&0&0&...&0\\0&2&0&...&0\\..&..&..&..&..\\0&0&...&0&2\end{pmatrix}$$

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