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COMEDK2020MathematicsDeterminants

The value of | array lll x & p & q p & x & q p & q & x array | is

Options

  1. A(x-p)(x-q)(x+p+q)
  2. Bx(x-p)(x-q)
  3. Cp q(x-p)(x-q)
  4. D(p-q)(x-q)(x-p)

Correct answer

A. (x-p)(x-q)(x+p+q)

Step-by-step solution

We have [ array lll x & p & q p & x & q p & q & x array ] Applying C₁ C₁+C₂+C₃₃ , we get | array lll x+p+q & p & q x+p+q & x & q x+p+q & q & x array | Taking common (x+p+q) from C₁ , we get (x+p+q) | array lll 1 & p & q 1 & x & q 1 & q & k array | Applying R₁ R₁-R₂, R₂ R₂-R₃ , we get aligned &(x+p+q) | array ccc 0 & p-x & 0 0 & x-q & q-x 1 & q & x array | &=(x+p+q) 1[(p-x)(q-x)-0] &= (x-p)(x-q)(x+p+q) aligned

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