Suppose the matrix $A$ is a $2 \times 2$ non-zero matrix with entries in $\Bbb C$.
Which of the following statements must be true?
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$PAP^{-1}$ is a diagonal matrix for some invertible matrix $P$ with entries in $\Bbb R$.
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$A$ has only one distinct eigenvalue in $\Bbb C$ with multiplicity $2$.
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$A$ has two distinct eigenvalues in $\Bbb C$.
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$Av = v$ for a non-zero $v$.
Please suggest which of the possibilities hold. It seems to me that the characteristic poly is $ f(t) = t^2$, which means option (2) holds that is only one eigenvalue zero with multiplicity $2$.
Best Answer
Hint: $\lambda$ eigenvalue of $A\Rightarrow \lambda^2$ eigenvalue of $A^2$. What are the eigenvalues of $A^2$, and what is hence the only possible eigenvalue of $A$? How can $A$ look like (up to basechange)?