Q1: Let $A \in \mathbb{R}^{3 \times 3}$ be a square matrix with $$ A = \begin{pmatrix} 2 & 0 & 0 \ 0 & 2 & 0 \ 0 & 0 & 2 \end{pmatrix} $$ What is algebraic multiplicity of the eigenvalue $2$?

A1: 3

A2: 0

A3: 1

A4: 2

Q2: Let $A \in \mathbb{R}^{n \times n}$ be a square matrix with eigenvalue $2$. What is never possible for the algebraic multiplicity $\alpha(2)$?

A1: $\alpha(2) = n+1$

A2: $\alpha(2) = n$

A3: $\alpha(2) = 1$

A4: $\alpha(2) \in \mathbb{R}$

Q3: Let $A \in \mathbb{R}^{2 \times 2}$ with eigenvalue $5$ and $7$. Which of the following claims for the algebraic multiplicity is the only possible one?

A1: $\alpha(5) = \alpha(7) = 1$

A2: $\alpha(5) + \alpha(7) = 1$

A3: $\alpha(5) + \alpha(7) = 3$

A4: $\alpha(5) = 2$

A5: $\alpha(7) = 0$