Aptitude | GATE CS 1998 | Question 37

Consider the following determinant


Which of the following is a factor of Δ ?
(A) a+b
(B) a-b
(C) a+b+c
(D) abc

Answer: (B)

Explanation:  \begin{vmatrix} 1&a&bc\\ 1&b&ac\\ 1&c&ab\\ \end{vmatrix} = \begin{vmatrix} 0&a-b&c(b-a)\\ 1&b&ac\\ 0&c-b&a(b-c)\\ \end{vmatrix} \\\\ \text{Taking }(a-b)\text{ common from R}_1 \text{ and }(c-b)\text{ from R}_3\\ = (a-b)(c-b) \begin{vmatrix} 0&1&-c\\ 1&b&ac\\ 0&1&-a\\ \end{vmatrix}
(a-b) is a factor of the determinant.
Therefore option (B) is correct.

This explanation is provided by Chirag Manwani.

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