Question

What is the value of determinant given if x, y, z are in AP with common difference d.

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

• A. $2d^2$
• B. $-d^2$
• C. $-3d^2$
• D. $d^2$

Answer 1

The correct option is C

$-3d^2$

You can use the property of the determinant that if the terms of one row is given as a sum of two other terms then the determinant can be written as the sum of two determinants as given below,

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

$=\left|\begin{array}{ccc}x& y& 1\\ y& z& 1\\ z& x& 1\end{array}\right|+0$

$R_2\to R_2-R_1$
$R_3\to R_3-R_1$

$=\left|\begin{array}{ccc}x& y& 1\\ y-x& z-y& 0\\ z-x& x-y& 0\end{array}\right|$

$=\left|\begin{array}{ccc}x& y& 1\\ d& d& 0\\ 2d& -d& 0\end{array}\right|$

$=+1(-d^2-2d^2)$
$=-3d^2$