Question

Find the value of the determinant $\left[\begin{array}{ccc}101& 102& 103\\ 104& 105& 106\\ 107& 108& 109\end{array}\right]$

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{\Delta }=\left|101102103 \\ 104105106 \\ 107108109\right| \\ \mathrm{\Delta }=\left|10112 \\ 10412 \\ 10712\right|\left[\mathrm{Applying}{\mathrm{C}}_2\to {\mathrm{C}}_2-{\mathrm{C}}_1\mathrm{and}{\mathrm{C}}_3\to {\mathrm{C}}_3-{\mathrm{C}}_1\right] \\ =2\left|10111 \\ 10411 \\ 10711\right| \\ =0 \\ \mathrm{Since}\mathrm{two}\mathrm{columns}\mathrm{are}\mathrm{identitical},\mathrm{the}\mathrm{value}\mathrm{of}\mathrm{the}\mathrm{determinant}\mathrm{is}\mathrm{zero}. \\ \Rightarrow \mathrm{\Delta }=\left|101102103 \\ 104105106 \\ 107108109\right|=0 \end{gathered}$$