Question

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

(a) $9x^2(x+y)$
(b) $9y^2(x+y)$
(c) $3y^2(x+y)$
(d) $7x^2(x+y)$

Answer 1

(b) We have, $\left|\begin{array}{ccc}x& x+y& x+2y\\ x+2y& x& x+y\\ x+y& x+2y& x\end{array}\right|$
$=\left|\begin{array}{ccc}3(x+y)& x+y& y\\ 3(x+y)& x& y\\ 3(x+y)& x+2y& -2y\end{array}\right|$ $[\because C_1\to C_1+C_2+C_3\text{and}{C}_3\to C_3-C_2]$
$=3(x+y)\left|\begin{array}{ccc}1& (x+y)& y\\ 1& x& y\\ 1& (x+2y)& -2y\end{array}\right|$ [taking 3(x+y) common from first column]
$=3(x+y)\left|\begin{array}{ccc}0& y& 0\\ 1& x& y\\ 1& (x+2y)& -2y\end{array}\right|$ $[\because R_1\to R_1-R_2]$
Expanding along $R_1$
$=3(x+y)[-y(-2y-y\left)\right]=3y^2.3(x+y)=9y^2(x+y)$