Question

Find the value of the determinant \(\begin{vmatrix}x&a&x+a\\y&b&y+b\\z&c&z+c\end{vmatrix}\).

Hint:

Column 3 is exactly column 1 + column 2 in every row, so the columns are linearly dependent.

Answer 1

Step 1: Recognize linear dependence directly:
Since $C_3=C_1+C_2$ (true entry by entry: $x+a$, $y+b$, $z+c$), the three columns are linearly dependent.

Step 2: A determinant vanishes whenever its columns are linearly dependent:
This is a standard property — no row reduction is even needed once the dependence is spotted.

Final Answer:
\[ \boxed{0} \]