Question

Using properties of determinants prove the following:
$\left|\begin{array}{ccc}x& x+y& x+2y\\ x+2y& x& x+y\\ x+y& x+2y& x\end{array}\right|=9y^2(x+y).$

Answer 1

Take L.H.S

$\left[\begin{array}{ccc}x& x+y& x+2y\\ x+2y& x& x+y\\ x+y& x+2y& x\end{array}\right]$

Now apply $C_1\to C_1+C_2+C_3$

$\left[\begin{array}{ccc}3x+3y& x+y& x+2y\\ 3x+3y& x& x+y\\ 3x+3y& x+2y& x\end{array}\right]$

Taking common $3(x+y)$ from $C_1$

$3(x+y)\left[\begin{array}{ccc}1& x+y& x+2y\\ 1& x& x+y\\ 1& x+2y& x\end{array}\right]$

Now apply, $R_1\to R_1-R_3$

$R_2\to R_2-R_3$

$3(x+y)\left[\begin{array}{ccc}0& -y& 2y\\ 0& -2y& x+y\\ 1& x+2y& x\end{array}\right]$

$3(x+y)\left[\begin{array}{cc}-y& 2y\\ -2y& y\end{array}\right]$

$3y^2(x+y)\left[\begin{array}{cc}-1& 2\\ 2& 1\end{array}\right]$

$3y^2(x+y)(-1+4)=9y^2(x+y)$

Hence proved.