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

Question

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?

Answer 1

Proof that (1) is false:

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

Answer 2

You are correct about 2 and 3.

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