Q1: Let $A \in \mathbb{C}^{n \times m}$ but each entry is actually a real number. What is always correct?

A1: $ A^* = A^T $

A2: $ A^* = A $

A3: $ A^* = - A^T $

A4: $ A^* = - A $

Q2: Consider the vector space $\mathbb{C}^2$ with the standard inner product $\langle \cdot, \cdot \rangle$. What is correct?

A1: $ \langle x, y \rangle = x^\ast y $

A2: $ \langle x, y \rangle = x^T y $

A3: $ \langle x, y \rangle = x y $

A4: $ \langle x, A y \rangle = x A^\ast y $

A5: $ \langle x, y \rangle = y -x $

Q3: Consider the a matrix $A \in \mathbb{C}^{n \times n}$ with spectrum given by $i, 1, 2, e^i$. What is the spectrum of the adjoint $A^\ast$?

A1: ${ - i, 1, 2, e^{-i} }$

A2: ${ 1, 2 }$

A3: ${ i, 1, 2, e^{i} }$

A4: ${ 0 }$