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Question
The value of determinant `|("a" - "b", "b" + "c", "a"),("b" - "a", "c" + "a", "b"),("c" - "a", "a" + "b", "c")|` is ______.
Options
a3 + b3 + c3
3bc
a3 + b3 + c3 – 3abc
None of these
Solution
The value of determinant `|("a" - "b", "b" + "c", "a"),("b" - "a", "c" + "a", "b"),("c" - "a", "a" + "b", "c")|` is none of these.
Explanation:
Here, we have `|("a" - "b", "b" + "c", "a"),("b" - "a", "c" + "a", "b"),("c" - "a", "a" + "b", "c")|`
C2 → C2 + C3
⇒ `|("a" - "b", "a" + "b" + "c", "a"),("b" - "a", "a" + "b" + "c", "b"),("c" - "a", "a" + "b" + "c", "c")|`
⇒ `("a" + "b" + "c") |("a" - "b", 1, "a"),("b" - "a", 1, "b"),("c" - "a", 1, "c")|` .....(Taking a + b + c common from C2)
R1 → R1 – R2, R2 → R2 – R3
⇒ `("a" + "b" + "c") |(2("a" - "b"), 0, "a" - "b"),("b" - "c", 0, "b" - "c"),("c" - "a", 1, "c")|`
Taking (a – b) and (b – c) common from R1 and R2 respectively
⇒ `("a" + "b" + "c")("a" - "b")("b" - "c") |(2, 0, 1),(1, 0, 1),("c" - "a", 1, "c")|`
Expanding along C2
⇒ `("a" + "b" + "c")("a" - "b")("b" - "c") [-1|(2, 1),(1, 1)|]`
⇒ (a + b + c)(a – b)(b – c)(– 1)
⇒ (a + b + c)(a – b)(c – b)
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