Question

यदि A = `[(1,sintheta,1),(-sintheta,1,sintheta),(-1,-sintheta,1)]`, जहाँ 0 ≤ θ ≤ 2π हो तो _____।

विकल्प

• det (A) = 0

• det (A) ∈ (2, ∞)

• det (A) ∈ (2, 4)

• det (A) ∈ [2, 4]

Answer 1

यदि A = `[(1,sintheta,1),(-sintheta,1,sintheta),(-1,-sintheta,1)]`, जहाँ 0 ≤ θ ≤ 2π हो तो det (A) ∈ [2, 4]।

स्पष्टीकरण:

माना A = `[(1,sintheta,1),(-sintheta,1,sintheta),(-1,-sintheta,1)]`

∴ |A| =  `[(1,sintheta,1),(-sintheta,1,sintheta),(-1,-sintheta,1)]`

= 1[1 + sin² θ] − sin θ[−sin θ + sin θ] + 1[sin² θ + 1]

= 1 + sin² θ + sin² θ + 1

= 2 + 2 sin² θ

= 2(1 + sin² θ)

जब θ = 0, π, 2π तो sin θ = 0

⇒ |A| = 2

जब θ = `pi/2, (3pi)/2` तो sin² θ = 1

|A| = 2(1 + 1)

= 2 × 2

= 4

∴ det A ∈ [2, 4]