Let $\chi^{2}$ denote the space of all prime sense double gai sequences and $\Lambda^{2}$ the space of all prime sense double analytic sequences. First we show that the set $E = \{s^{(mn)}: m,n = 1, 2, 3, \dots\}$ is a determining set for $\chi_{\pi}^{2}$. The set of all finite matrices transforming $\chi_{\pi}^{2}$ into FK-space $Y$ denoted by $(\chi_{\pi}^{2}: Y)$. We characterize the classes $(\chi_{\pi}^{2}: Y)$ when $Y = c_{0}^{2}, c^{2}, \chi^{2}, l^{2}, \Lambda^{2}$. But the approach to obtain these result in the present paper is by determining set for $\chi_{\pi}^{2}$. First, we investigate a determining set for $\chi_{\pi}^{2}$ and then we characterize the classes of matrix transformations involving $\chi_{\pi}^{2}$ and other known sequence spaces.