Question

Using the property of determinants and without expanding, prove that
$\left|\begin{array}{ccc}x& a& x+a\\ y& b& y+b\\ z& c& z+c\end{array}\right|=0$

Answer 1

Let $\Delta =\left|\begin{array}{ccc}x& a& x+a\\ y& b& y+b\\ z& c& z+c\end{array}\right|$
Applying $C_2=C_2+C_1$ we get,
$\Delta =\left|\begin{array}{ccc}x& x+a& x+a\\ y& y+b& y+b\\ z& z+c& z+c\end{array}\right|=0$
Since second and third collumn are same.