How can a matrix be Hermitian, unitary, and diagonal all at once?

Hint : Here I have done for $2 \times 2$ matrix.

Let $A = \left( \begin{array}{cc} a & 0 \\ 0 & b \\ \end{array} \right)$

be a diagonal matrix with complex entries. Its eigenvalues are precisely $a$, $b$. Because $A$ is Hermitian, they must be real. Also $A$ is unitary, they must each be of absolute value $1$. There are exactly four matrices satisfying these conditions:

Let $A_1 = \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array} \right)$, $A_2 = \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \\ \end{array} \right)$, $A_3 = \left( \begin{array}{cc} -1 & 0 \\ 0 & 1 \\ \end{array} \right)$, $A_4 = \left( \begin{array}{cc} -1 & 0 \\ 0 & -1 \\ \end{array} \right)$

I hope this may help you.

Start with the definitions of Hermitian, unitary, and diagonal. You should start with the definition of diagonal...