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NDA 2018 Maths · Mathematics · Matrices · hard

What is the value of the determinant beginvmatrix1&a&b+c\\1&b&c+a\\1&c&a+bendvmatrix?

Single correct option

What is the value of the determinant ∣1ab+c1bc+a1ca+b∣\begin{vmatrix}1&a&b+c\\1&b&c+a\\1&c&a+b\end{vmatrix}?
  1. A. 0
  2. B. a+b+ca+b+c
  3. C. abcabc
  4. D. (a−b)(b−c)(c−a)(a-b)(b-c)(c-a)
Show answer and solution

Answer: (A) 0

Add the second column to the third: C3→C3+C2C_3\to C_3+C_2 gives entries a+b+ca+b+c in every row. Then column 3 is (a+b+c)×(a+b+c)\times column 1, so two columns are proportional and the determinant is 0.
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Original question written in the NDA 2018 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake