Q1: Let $A \in \mathbb{R}^{n \times n}$ be a square matrix. What is the correct definition for the characteristic polynomial?

A1: $\lambda \mapsto \det(A - \lambda \mathbf{1} )$

A2: $\lambda \mapsto \det(A + \lambda \mathbf{1} )$

A3: $\lambda \mapsto \det( \lambda A - \mathbf{1} )$

A4: $\lambda \mapsto \det( \lambda A + \mathbf{1} )$

Q2: Let $A \in \mathbb{R}^{n \times n}$ be a square matrix. What is not correct for the characteristic polynomial $p_A(\lambda) = \det(A - \lambda \mathbf{1} )$?

A1: The leading coefficient of $p_A$ is always 1.

A2: $p_A$ is a polynomial of degree $n$

A3: $p_A(0) = \det(A)$

A4: $p_A(\lambda) = (-1)^n \lambda^n + \cdots + c_1 \lambda + c_0 $

Q3: Let $A \in \mathbb{R}^{2 \times 2}$ be a square matrix given by $\begin{pmatrix} 2 & 1 \ 1 & 3 \end{pmatrix}$. What is the characteristic polynomial $p_A(\lambda) = \det(A - \lambda \mathbf{1} )$ in this case?

A1: $p_A(\lambda) = (2 - \lambda )(3 - \lambda ) - 1$

A2: $p_A(\lambda) = (2 - \lambda )(3 - \lambda ) + 1$

A3: $p_A(\lambda) = (2 - \lambda )(3 - \lambda ) $

A4: $p_A(\lambda) = (2 - \lambda )(3 - \lambda ) - (1-\lambda)(1-\lambda)$

A5: $p_A(\lambda) = (2 - \lambda )(1 - \lambda ) - 3$