$\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}x&a&{x + a}\\y&b&{y + b}\\z&c&{z + c}\end{array}} \right| = 0$

Solution

.: L.H.S. $= \left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}x&a&{x + a}\\y&b&{y + b}\\z&c&{z + c}\end{array}} \right|$

$= \left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}x&a&x\\y&b&y\\z&c&z\end{array}} \right| + \left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}x&a&a\\y&b&b\\z&c&c\end{array}} \right| = 0$

[If any two rows or columns of a determinant are identical

(all corresponding elements are same), the value of determinant is zero].