Question

The value of 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. $9x^2(x+y)$
• B. $9y^2(x+y)$
• C. $3y^2(x+y)$
• D. $7x^2(x+y)$

Answer 1

The correct option is B $9y^2(x+y)$

We have the given determinant

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

$C_1\to C_1+C_2+C_3$

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

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

$C_3\to C_3-C_2$

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

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

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

$3y\left(x+y\right)\left[3\left\{\left(x+y\right)-x\right\}\right]$

$=3y(x+y)3y$

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

Hence the value of given determinants is$=9y^2(x+y)$