Evaluate the determinant
\(\begin{vmatrix} x-y & y-z & z-x\\y-z & z-x & x-y\\z-x & x-y & y-z \end{vmatrix}\)
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 0
In every row the three elements add up to zero:
\((x - y) + (y - z) + (z - x) = 0\)
Apply \(C_1 \to C_1 + C_2 + C_3\), which does not change the value of the determinant:
\( \Delta = \begin{vmatrix} 0 & y-z & z-x\\0 & z-x & x-y\\0 & x-y & y-z \end{vmatrix} \)
A determinant with a column of zeros is 0, so \(\Delta = 0\).