Consider DFT matrix for N=4,$A = \frac{1}{\sqrt{4}} \begin{bmatrix} \ 1 & 1 & 1 &1 \\ \ 1 & -j & -1 & j \\ \ 1 & -1 & 1 & -1 \\ \ 1 & j & -1 & -j \\ \end{bmatrix}$

If AA*=I

Then A is a unitary matrix

For N=4,

$AA * = \frac{1}{\sqrt{4}} \begin{bmatrix} \ 1 & 1 & 1 &1 \\ \ 1 & -j & -1 & j \\ \ 1 & -1 & 1 & -1 \\ \ 1 & j & -1 & -j \\ \end{bmatrix} \frac{1}{\sqrt{4}} \begin{bmatrix} \ 1 & 1 & 1 &1 \\ \ 1 & j & -1 & -j \\ \ 1 & -1 & 1 & -1 \\ \ 1 & -j & -1 & j \\ \end{bmatrix}$

$\hspace{1.0cm}= \frac{1}{4} \begin{bmatrix} \ 4 & 0 & 0 & 0 \\ \ 0 & 4 & 0 & 0 \\ \ 0 & 0 & 4 & 0 \\ \ 0 & 0 & 0 & 4 \\ \end{bmatrix}$

$\hspace{1.0cm}= \begin{bmatrix} \ 1 & 0 & 0 & 0 \\ \ 0 & 1 & 0 & 0 \\ \ 0 & 0 & 1 & 0 \\ \ 0 & 0 & 0 & 1 \\ \end{bmatrix}$

AA*= I

Therefore DFT matrix A is a unitary matrix.