Q183 / 270 · NDA 2017 (Phase 1) · Mathematics

What is the value of the determinant? $\left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & 1+\mathrm{xyz} & 1 \\ 1 & 1 & 1+\mathrm{xyz}\end{array}\right| ?$

  • A
    $1+x+y+z$
  • B
    $2 x y z$
  • Correct answer
    C
    $x^2 y^2 z^2$
  • D
    $2 x^2 y^2 z^2$

Solution

$\begin{aligned} & \left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & 1+\mathrm{xyz} & 1 \\ 1 & 1 & 1+\mathrm{xyz}\end{array}\right| \quad \begin{array}{c}c_2 \rightarrow c_2-c_1 \\ c_3 \rightarrow c_3-c_1\end{array} \\ & =\left|\begin{array}{ccc}1 & 0 & 0 \\ 1 & x y z & 0 \\ 1 & 0 & x y z\end{array}\right|=1\left(x^2 y^2 z^2-0\right)=x^2 y^2 z^2\end{aligned}$

Hence, the correct answer is option: 'C'.

Relevant Questions