The value of \(\begin{vmatrix}x&x + y&x + y + z\\2x&3x+2y&4x+3y+2z\\3x& 6x+3y& 10x+6y+3z\end{vmatrix}\) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(x^3\)
Using the row operations \(R_2 \to R_2 - 2R_1\) and \(R_3 \to R_3 - 3R_1\):
\( \displaystyle \Delta = \begin{vmatrix} x & x+y & x+y+z \\ 0 & x & 2x+y \\ 0 & 3x & 7x+3y \end{vmatrix} \)
Then \(R_3 \to R_3 - 3R_2\):
\( \displaystyle \Delta = \begin{vmatrix} x & x+y & x+y+z \\ 0 & x & 2x+y \\ 0 & 0 & x \end{vmatrix} \)
The determinant is now upper triangular, so it equals the product of its diagonal entries:
\( \Delta = x \cdot x \cdot x = x^3 \)