Question

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

Answer 1

Adding $R_1\to R_1+R_2+R_3$

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

Taking (3x+3y) common, We get

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

$C_1\to C_1-C_2$

$C_2\to C_2-C_3$

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

Finding out determinant We get,

= $(3x+3y)\left(3y^2\right)$

= $9y^2(3x+3y)$