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NDA2013MathematicsDeterminantsActual

What is the value of the determinant | array lll 1 & b c & a(b+c) 1 & c a & b(c+a) 1 & a b & c(a+b) array | ?

Options

  1. A0
  2. Babc
  3. Cab +bc +ca
  4. Dabc (a +b +c)

Correct answer

A. 0

Step-by-step solution

aligned & | array lll 1 & b c & a(b+c) 1 & c a & b(c+a) 1 & a b & c(a+b) array | & = | array lll 1 & b c & a b+b c+c a 1 & c a & b c+a b+c a 1 & a b & a c+b c+a b array | (Applying C₃=C₃+C₂) & =(a b+b c+c a) | array lll 1 & b c & 1 1 & c a & 1 1 & a b & 1 array | & =(a b+b c+c a) | array lll 1 & b c & 0 1 & c a & 0 1 & a b & 0 array | (Applying C₃=C₃-C₁) & =(a b+b c+c a)=0 aligned

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