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NDA 2017
Exam
NDA
Year
2017
Subject
Mathematics
Chapter
Determinants

NDA 2017 · Mathematics · Determinants

What is the value of the determinant? | array ccc 1 & 1 & 1 1 &...

What is the value of the determinant? \left|11111+xyz1111+xyz\begin{array}{ccc}1 & 1 & 1 \\ 1 & 1+\mathrm{xyz} & 1 \\ 1 & 1 & 1+\mathrm{xyz}\end{array}\right| ?

Options

  1. A1+x+y+z1+x+y+z
  2. B2xyz2 x y z
  3. Cx2y2z2x^2 y^2 z^2
  4. D2x2y2z22 x^2 y^2 z^2

Correct answer

C. x2y2z2x^2 y^2 z^2

Step-by-step solution

∣11111+xyz1111+xyz∣c2→c2−c1c3→c3−c1=∣1001xyz010xyz∣=1(x2y2z2−0)=x2y2z2\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}
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