Question

The value of the determinant $∆=\left|\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 3x& 6x& 9x\end{array}\right|$ is ____________.

Answer 1

Given: $∆=\left|\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 3x& 6x& 9x\end{array}\right|$

$$\begin{gathered} ∆=\left|\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 3x& 6x& 9x\end{array}\right| \\ \mathrm{Taking}\left(3x\right)\mathrm{common}\mathrm{from}{R}_3 \\ \Rightarrow ∆=\left(3x\right)\left|\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 1& 2& 3\end{array}\right| \\ \Rightarrow ∆=\left(3x\right)\left(0\right)\left(\because \mathrm{The}\mathrm{value}\mathrm{of}\mathrm{determinant}\mathrm{with}\mathrm{two}\mathrm{identicals}\mathrm{rows}\mathrm{is}\mathrm{equal}\mathrm{to}\mathrm{zero}\right) \\ \Rightarrow ∆=0 \end{gathered}$$

Hence, the value of the determinant $∆=\left|\begin{array}{ccc}1& 2& 3\\ 4& 5& 6\\ 3x& 6x& 9x\end{array}\right|$ is 0.