Q1: Let $A \in \mathbb{R}^{n \times n}$ be a square matrix. Is there a vector such that the equation $A \mathbf{x} = 3 \mathbf{x}$ is satisfied?

A1: Yes, there is at least one.

A2: No, there is never such a vector.

A3: Yes, there is always excatly one vector for this.

A4: One needs more information.

Q2: Let $A \in \mathbb{R}^{n \times n}$ be a square matrix. If there is a vector $\mathbf{x} \neq 0$ such that the equation $A \mathbf{x} = 3 \mathbf{x}$ is satisfied, what are to correct names?

A1: $3$ is called an eigenvalue of $A$ and $x$ an eigenvector.

A2: $3$ is called an eigenvector of $A$ and $x$ an eigenvalue.

A3: $3$ is called an eigenspace of $A$ and $x$ an eigenvalue.

A4: $3$ is called a scalar of $A$ and $x$ an eigenvalue.

Q3: Let $A \in \mathbb{R}^{2 \times 2}$ be a square matrix given by $\begin{pmatrix} 2 & 1 \ 0 & 3 \end{pmatrix}$. Try finding the eigenvaluas similarly to the procedure in the video. What is the spectrum of $A$?

A1: ${ 2,3 }$

A2: ${ 1, 2,3 }$

A3: ${ 0, 1, 2,3 }$

A4: ${ 1}$

A5: ${ 0 }$