Question

Prove that: `|("a"^2 + 2"a", 2"a" + 1, 1),(2"a" + 1, "a" + 2, 1),(3, 3, 1)| = ("a" - 1)^3`

Answer 1

 `|("a"^2 + 2"a", 2"a" + 1, 1),(2"a" + 1, "a" + 2, 1),(3, 3, 1)|`

[Applying R₁ → R₁ – R₂ and R₂ → R₂ – R₃]

=  `|("a"^2 - 1, "a" - 1, 0),(2"a" - 2, "a" - 1, 0),(3, 3, 1)|`

[Taking (a – 1) common from R₁ and R₂]

`("a" - 1)^2 |("a" + 1, 1, 0),(2, 1, 0),(3, 3, 1)|`

[Expanding along R₃]

= `("a" - 1)^2 [1 * ("a" + 1) - 2]`

= (a – 1)³