The value of the determinant \(\begin{vmatrix}1 & a & b + c\\ 1 & b & c + a\\ 1 & c & a + b\end{vmatrix}\) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 0
Adding one column to another does not change the value of a determinant. Apply \(C_3 \to C_3 + C_2\):
\( \displaystyle \Delta = \begin{vmatrix}1 & a & a + b + c\\ 1 & b & a + b + c\\ 1 & c & a + b + c\end{vmatrix} \)
Take the common factor \(a + b + c\) out of the third column:
\( \displaystyle \Delta = (a + b + c)\begin{vmatrix}1 & a & 1\\ 1 & b & 1\\ 1 & c & 1\end{vmatrix} \)
The first and third columns are now identical. A determinant with two identical columns is zero, so
\( \Delta = 0 \)