Unitary and hermitian matrices

Which of the following linear transformations $T:\mathbb{C}^{2}\to\mathbb{C}^{2}$ is unitary?

1. $T\left(\left[\begin{array}{c}x\\ y\end{array}\right]\right)=\left[\begin{array}{c}x\\ iy\end{array}\right]$.

2. $T\left(\left[\begin{array}{c}x\\ y\end{array}\right]\right)=\left[\begin{array}{c}iy\\ -ix\end{array}\right]$.

3. $T\left(\left[\begin{array}{c}x\\ y\end{array}\right]\right)=\left[\begin{array}{c}x+iy\\ ix+y\\ \end{array}\right]$.

4. $T\left(\left[\begin{array}{c}x\\ y\end{array}\right]\right)=\left[\begin{array}{c}\frac{1}{\sqrt{2}}x+\frac{i}{\sqrt{2}}y\\ \frac{1}{\sqrt{2}}x-\frac{i}{\sqrt{2}}y\\ \end{array}\right]$.

Determine which of the following matrices are unitary.

Determine which of the following matrices are unitary.

Assume $A$ and $B$ are unitary $3\times 3$-matrices. Then which of the following matrices are unitary?

Assume $A$ and $B$ are $3\times 3$-matrices, and assume $A$ is unitary and $B$ is invertible. Then which of the following matrices are unitary?

Assume $A$ and $B$ are hermitian $3\times 3$-matrices, and $A$ is invertible. Then which of the following matrices are hermitian?

Unitarily diagonalize the hermitian matrix $A=\left[\begin{array}{cc}2&1+i\\ 1-i&3\\ \end{array}\right]$.