Question

The value of the determinant $\left|\begin{array}{ccc}x& x+y& x+2y\\ x+2y& x& x+y\\ x+y& x+2y& x\end{array}\right|$ is

(a) 9x²(x + y)
(b) 9y²(x + y)
(c) 3y²(x + y)
(d) 7x²(x + y)

Answer 1

$$\begin{gathered} \left|\begin{array}{ccc}x& x+y& x+2y\\ x+2y& x& x+y\\ x+y& x+2y& x\end{array}\right| \\ =\left|\begin{array}{ccc}-2y& y& y\\ x+2y& x& x+y\\ -y& 2y& -y\end{array}\right|\left[\mathrm{Applying}{R}_1\to R_1-R_2\mathrm{and}{R}_3\to R_3-R_2\right] \\ =y^2\left|\begin{array}{ccc}-2& 1& 1\\ x+2y& x& x+y\\ -1& 2& -1\end{array}\right|\left[\mathrm{Taking}\left(y\right)\mathrm{common}\mathrm{from}{R}_1\mathrm{and}\mathrm{from}{R}_3\right] \\ =y^2\left|\begin{array}{ccc}-2& -3& 3\\ x+2y& 3x+4y& -y\\ -1& 0& 0\end{array}\right|\left[\mathrm{Applying}{C}_2\to C_2+2C_1\mathrm{and}{C}_3\to C_3-C_1\right] \\ =y^2\left[-1\left(3y-9x-12y\right)\right] \\ =y^2\left[9y+9x\right] \\ =9y^2\left(y+x\right) \end{gathered}$$

Hence, the correct option is (b).