Problem 10
The conjugate of a complex number $z = a + bi$ is $a - bi$. We denote the conjugate by $\bar{z}$.
We also define the modulus of a complex number $z$, denoted by $|z|$, as $\sqrt{z\bar{z}}$. Note that $z\bar{z}$ is always
a nonnegative real number, so this is also a nonnegative real number.
Let A be a complex $n \times n$ matrix, where $n \ge 2$. We define the matrix $\bar{A}$ to be the matrix obtained
by taking the conjugate of every entry of A.
a. Calculate $\bar{A}$, where $A = \begin{bmatrix} 2 - i & 3 + 2i \\ 4 & -i \end{bmatrix}$.
b. Calculate $|2 - 4i|$.
c. Prove that for any complex numbers $x$ and $y$, $\overline{(xy)} = \overline{(x)}\overline{(y)}$ and $\overline{x + y} = \overline{x} + \overline{y}$.
d. Prove that $|yz| = |y||z|$ for any $y, z \in \mathbb{C}$.
e. Prove that $det \bar{A} = \overline{det A}$. (Hint: Show the $2 \times 2$ case and use induction on $n$.)
f. A unitary matrix $A \in M_{n \times n}(\mathbb{C})$ is one where $(A^T) = A^{-1}$. Prove that if A is a unitary matrix,
then $|det A| = 1$.
