Question

Using the property of determinants and without expanding, prove that: $\begin{vmatrix}x&a&x+a\\\y&b&y+b\\\z&C&z+c\end{vmatrix}=0$

Answer 1

$\begin{vmatrix}x&a&x+a\\\y&b&y+b\\\z&C&z+c\end{vmatrix}$

=$\begin{vmatrix}x&a&x\\\y&b&y\\\z&c&z\end{vmatrix}+\begin{vmatrix}x&a&a\\\y&b&b\\\z&c&c\end{vmatrix}=0+0=0$

[Here the two columns of the determinants are identical]