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 geometric multiplicity of the eigenvalue $2$?

A1: 3

A2: 0

A3: 1

A4: 2

Q2: Let $A \in \mathbb{R}^{3 \times 3}$ be a square matrix with $$ A = \begin{pmatrix} 2 & 1 & 1 \ 0 & 2 & 1 \ 0 & 0 & 2 \end{pmatrix} $$ What is geometric multiplicity of the eigenvalue $2$?

A1: 1

A2: 0

A3: 3

A4: 2

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

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

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

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

A4: $\gamma(5) = 2$

A5: $\gamma(7) = 0$