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Consider the following determinant $\Delta = \begin{vmatrix} 1 & a & bc \\ 1 & b & ca \\ 1 & c & ab \end{vmatrix}$

Which of the following is a factor of $\Delta$?

  1. $a+b$
  2. $a-b$
  3. $a+b+c$
  4. $abc$
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Best answer
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$\begin{vmatrix} 1 &a &bc \\ 1& b& ca\\ 1& c& ab \end{vmatrix}$

$\begin{vmatrix} 1 &a &bc \\ 0& b-a& ca-bc\\ 0& c-a& ab-bc \end{vmatrix} \ \ \ \ \ \ \ \ R_{2}\rightarrow R_{2}- R_{1} \ \ , \ R_{3}\rightarrow R_{3}- R_{1}$

$(b-a)( ab-bc) - (c-a)(ca-bc)$

$-(a-b) \ \ b \ \ ( a-c) + (a-c)\ \ c \ \ (a-b)$

$(a-b)( a-c) (c-b)$

Option$(B)a-b$ is the correct choice.
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Answer is B

R2->R2 - R1

R3 -> R3 - R2

you will gt det = (a-b)*(a-c)*(b+c)

in matrix operations, you cannot multiply rows or columns. That will not yield the same matrix. So abc is not correct
Answer:

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