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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
Solution
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]
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