If \(x, y\) and \(z\) are real numbers such that \(x + y + z = 0\), then value of \(\begin{vmatrix}3x&-x+y&-x+z\\x-y&3y&z-y\\x-z&y-z&3z\end{vmatrix}\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 0
The operation \(C_1 \rightarrow C_1 + C_2 + C_3\) does not change the value of a determinant. The new first column holds the row sums:
\(3x + (-x + y) + (-x + z) = x + y + z\)
\((x - y) + 3y + (z - y) = x + y + z\)
\((x - z) + (y - z) + 3z = x + y + z\)
Since \(x + y + z = 0\), every entry of the new first column is 0. A determinant with a column of zeros is 0.