We have a matrix
$$\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}$$
This matrix represents a transformation in homogeneous coordinates. My question is whether the above matrix is affine or not and an example too for this.
It is not necessarily affine. An affine matrix in homogeneous coordinates has a form like:
$$\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ 0 & 0 & 1 \end{bmatrix}$$
(assuming you use a column vector convention). Here, the upper-left 2×2 submatrix is the linear part, and $(a_{13}, a_{23})$ is the translation vector of the affine transform.
If the lower row of the matrix has some values other than $[0, 0, 1]$, then it is in general a projective transform, not an affine one.