Let M = sin 4 ⁡ θ - 1 - sin 2 ⁡ θ 1 + cos 2 ⁡ θ cos 4 ⁡ θ = α I + β M - 1 , where α = α θ and β = β θ are real number, and I is the 2 × 2 identity matrix. If α * is the minimum of the set α θ : θ ∈ 0 ,   2 π and β * is the minimum of the set β θ : θ ∈ 0 ,   2 &#96

Step-by-step solution

Method I As given, M = s i n 4 θ - 1 - s i n 2 θ 1 + c o s 2 θ c o s 4 θ = α I + β M - 1 a d j   M = c o s 4 θ 1 + s i n 2 θ - 1 - c o s 2 θ s i n 4 θ M - 1 = 1 | M | c o s 4 θ 1 + s i n 2 θ - 1 - c o s 2 θ s i n 4 θ s i n 4 θ - 1 - s i n 2 θ 1 + c o s 2 θ c o s 4 θ = α 1 0 0 1 + β M c o s 4 θ 1 + s i n 2 θ - 1 - c o s 4 θ s i n 4 θ Compare element a 12 in L.H.S. and R.H.S. - 1 - s i n 2 &#9