The present study introduces results about unique solvability of Gaussian RBF interpolation with the different data sites and basis centers. For , we show that the interpolation matrix is singular only when the vector of difference between basis centers and the vector of difference between data sites are perpendicular to each other. For , we show certain states that the interpolation matrix is singular, then we provide several mild conditions which guarantee the interpolation matrix to be non-singular. We propose an algorithm to describe how to choose the basis centers and data sites. The results show that if the basis centers are chosen different from the data sites, the interpolation is uniquely solvable under mild conditions.