\( \begin{vmatrix}x+1 & x & x\\x & x+1 & x\\x & x & x+1\end{vmatrix} \) equals
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( 3x+1 \)
Every row adds up to 3x + 1, so \( C_1\to C_1+C_2+C_3 \) lets us take 3x + 1 out of the first column:
\( \displaystyle \Delta=(3x+1)\begin{vmatrix}1 & x & x\\1 & x+1 & x\\1 & x & x+1\end{vmatrix} \)
With \( R_2\to R_2-R_1 \) and \( R_3\to R_3-R_1 \):
\( \displaystyle \Delta=(3x+1)\begin{vmatrix}1 & x & x\\0 & 1 & 0\\0 & 0 & 1\end{vmatrix} \)
\( \displaystyle =(3x+1)(1)=3x+1 \)
Shortcut: Put x = 0: the determinant becomes that of the identity matrix, which is 1. The options give 3, 1, 4 and −3, so only Option (2) fits.