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

Answer 1

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)