Question

The determinant `|("b"^2 - "ab", "b" - "c", "bc" - "ac"),("ab" - "a"^2, "a" - "b", "b"^2 - "ab"),("bc" - "ac", "c" - "a", "ab" - "a"^2)|` equals ______.

Options

• abc (b–c) (c – a) (a – b)

• (b–c) (c – a) (a – b)

• (a + b + c) (b – c) (c – a) (a – b)

• None of these

Answer 1

The determinant `|("b"^2 - "ab", "b" - "c", "bc" - "ac"),("ab" - "a"^2, "a" - "b", "b"^2 - "ab"),("bc" - "ac", "c" - "a", "ab" - "a"^2)|` equals none of these.

Explanation:

We have, `|("b"^2 - "ab", "b" - "c", "bc" - "ac"),("ab" - "a"^2, "a" - "b", "b"^2 - "ab"),("bc" - "ac", "c" - "a", "ab" - "a"^2)|` 

= `|("b"("b" - "a"), "b" - "c", "c"("b" - "a")),("a"("b" - "a"), "a" - "b", "b"("b" - "a")),("c"("b" - "a"), "c" - "a", "a"("b" - "a"))|`

[Taking (b – a) common from C₁ and C₃ each]

= `("b" - "a")^2 |("b", "b" - "c", "c"),("a", "a" - "b", "b"),("c", "c" - "a", "a")|`

[Applying C₂ → C₂ + C₃]

= `("b" - "a")^2 |("b", "b", "c"),("a", "a", "b"),("c", "c", "a")|`

= 0  .....[As C₁ and C₂ are identical]