If a_1, a_2, a_3, , a_9 are in GP, then what is the value of the...

As $a_1, a_2, a_3 \ldots \ldots a_9$ are in G.P. $\therefore \frac{a_4}{a_1}=\frac{a_5}{a_2}=\frac{a_6}{a_3} \ldots . .=r^3 \quad(\text{ where } r=\text{ common ratio) }$ Now, \left|$\begin{array}{lll}\ln a_1 & \ln a_2 & \ln a_3 \\ \ln a_4 & \ln a_5 & \ln a_6 \\ \ln a_7 & \ln a_8 & \ln a_9\end{array}$\right|=\left|\begin{array}{ccc}\ln a_4-\ln a_1 & \ln a_5-\ln a_2 & \ln a_6-\ln a_3 \\ \ln a_7-\ln a_4 & \ln a_8-\ln a_5 & \ln a_9-\ln a_6 \\ \ln a_7 & \ln a_8 & \ln a_9\end{array}\right|$ (Applying $R_1 \rightarrow R_2-R_1$ and $R_2 \rightarrow R_3-R_2$ ) $=\left|\begin{array}{lll}\ln\frac{a_4}{a_1} & \ln\frac{a_5}{a_2} & \ln\frac{a_6}{a_3} \\ \ln\frac{a_7}{a_4} & \ln\frac{a_8}{a_5} & \ln\frac{a_9}{a_6} \\ \ln a_7 & \ln a_8 & \ln a_9\end{array}\right|=\left|\begin{array}{lll}\ln r^3 & \ln r^3 & \ln r^3 \\ \ln r^3 & \ln r^3 & \ln r^3 \\ \ln a_7 & \ln a_8 & \ln a_9\end{array}$$\right|=0