Q1: Let $A \in \mathbb{C}^{n \times n}$ with $2,4,6, \ldots, 2n$ as eigenvalues. Is $A$ diagonalizable?

A1: Yes, it is.

A2: No, it’s never diagonalizable.

A3: One needs more information.

Q2: Let $A \in \mathbb{C}^{2 \times 2}$ where $1$ is the only eigenvalue. Assume $A$ is diagonalizable. What is also correct?

A1: The geometric multiplicity of the eigenvalue $1$ is exactly 2.

A2: The algebraic multiplicity of the eigenvalue $1$ is 1.

A3: $A$ has $1$ and $2$ on the diagonal.

Q3: Let $A \in \mathbb{C}^{3 \times 3}$ be not diagonalizable. What is a correct conclusion?

A1: $A$ is not normal.

A2: $A$ has $3$ different eigenvalues.

A3: $A$ has only one eigenvalue.

A4: There is no eigenspace with dimension $2$.