Let M = ⁡θ − 1; −sin 2⁡θ 1 + cos 2⁡θ cos 4⁡θ] = αI + βM⁻¹; Where α = α(θ) and β = β(θ) are real numbers and I is the 2 × 2 identity matrix. If α∗ = is the minimum of the set α(θ): θ ∈ [0, 2π] and β∗ = is the minimum of the set β(θ): θ ∈ [0, 2π), Then the value of α∗ + β∗ is

Correct option is B. −2;916
m = sin4θcos4θ + sin2θcos2θ2 + sin4θcos4θ + sin2θcos2θ[sin4 − (1 + sin2θ) 1 + cos2θ cos4θ] = [α00α] + β = 1|m|[cos4θ1 + sin−1; −cos2θsinsin4θ = α + β|m|cos4θ, −1; −sin2θ = β|m|(1 + sin2θ)β = −|m|β = −[sin4θcos4θ + sin2θcos2θ + 2] = −[t² + t + 2] ⇒ βmin = −α = sin4θ + cos4θ = 1 − 2;sin2θcos2θ = 1 − 1;2(sin22θ) ⇒ min α = 12α + β = −3716 + 12 = −3716 + 816 = −2;916.