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If a 2*2 matrix has only one independent eigen vector then which of the following is necessarily true?

  1. Inverse does not exist
  2. There must be a repeated eigen value
  3. The matrix is non diagonalizable

I am sure about 2 and 3. I believe the 1 is not right. Can anyone give an example?

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Proof that (1) is false:

$$A=\begin{pmatrix}1&1\\0&1\end{pmatrix}$$

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You are correct about 2 and 3.

To 1: $$\begin{pmatrix}1 & 1\\ 0 & 1\end{pmatrix}^{-1} = \begin{pmatrix} 1 & -1 \\ 0 & 1 \\ \end{pmatrix}. $$

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