Question

If `|("a", "b", "a"alpha + "b"),("b", "c", "b"alpha + "c"),("a"alpha + "b", "b"alpha + "c", 0)|` = 0, prove that a, b, c are in G. P or α is a root of ax² + 2bx + c = 0

Answer 1

Let Δ = `|("a", "b", "a"alpha + "b"),("b", "c", "b"alpha + "c"),("a"alpha + "b", "b"alpha + "c", 0)|` 

= `|("a", "b", "a"alpha),("b", "c", "b"alpha),("a"alpha + "b", "b"alpha + "c", -("b"alpha + c))|  ("C"_3 -> "C"_3 - "C"_2)`  

= `|("a", "b", 0),("b", "c", 0),("a"alpha + "b", "b"alpha + "c", -("b"alpha + c)),(, , -("a"alpha^2 + "b"alpha))|  ("C"_3 -> "C"_3 - alpha"C"_1)` 

= `|("a", "b", 0),("b", "c", 0),("a"alpha + "b", "b"alpha + "c", -("a"alpha^2 + 2"b"alpha + c))|` expanding along C₃

We get – (aα² + 2bα + c)[ac – b²]

So Δ = 0

⇒ (aα² + 2bα + c)(ac – b²)

= – 0

= 0

⇒ aα² + 2bα + c = 0

or

ac – b² = 0

(i.e.) a is a root of ax² + 2bx + c = 0

or

ac = b²

⇒ a, b, c are in G.P.