Q1: Let $A \in \mathbb{C}^{n \times n}$ be given by the identity matrix. What is not correct?

A1: $A$ is skew-adjoint.

A2: $ A^* = A $.

A3: $A$ is unitary.

A4: $A$ is normal.

A5: $A^{-1} = A^\ast$.

Q2: Let $A \in \mathbb{C}^{2 \times 2}$ be given by $ \begin{pmatrix} i & 0 \ 0 & i\end{pmatrix}$. What is not correct?

A1: $A$ is selfadjoint.

A2: $A$ is skew-adjoint.

A3: $A$ is unitary.

A4: $A$ is normal.

A5: $A^{-1} = A^\ast$.

Q3: Let $A \in \mathbb{C}^{n \times n}$. What can you definitely say about the matrix $B$ given by $$ B = \frac{1}{2 i} ( A - A^\ast)$$

A1: $B$ is selfadjoint.

A2: $B$ is skew-adjoint.

A3: $B$ is unitary.

A4: $B$ only has real numbers as entries.