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NDA 2013
Exam
NDA
Year
2013
Subject
Mathematics
Chapter
Determinants

NDA 2013 · Mathematics · Determinants

The value of the determinant | array ccc m & n & p p & m & n n & p...

The value of the determinant \left|mnppmnnpm\begin{array}{ccc}m & n & p \\ p & m & n \\ n & p & m\end{array}$$\right|

Options

  1. Ais a perfect cube
  2. Bis a perfect square
  3. Chas linear factor
  4. Dis zero

Correct answer

C. has linear factor

Step-by-step solution

(m+n+p)\left|1np1mn1pm\begin{array}{ccc}1 & n & p \\ 1 & m & n \\ 1 & p & m\end{array}\right|(m+n+p)\left|1np0m−nn−p0p−nm−p\begin{array}{ccc}1 & n & p \\ 0 & m-n & n-p \\ 0 & p-n & m-p\end{array}\right|=(m+n+p)(m2−mp−mn+np−np+p2+n2−np)=(m+n+p)(m2+n2+p2−mn−np−pm)\begin{aligned} & =(m+n+p)\left(m^2-m p-m n+n p-n p+p^2+n^2-n p\right) \\ & =(m+n+p)\left(m^2+n^2+p^2-m n-n p-p m\right)\end{aligned}$$
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