Let A = 4 - 2 α β . If A 2 + γ A + 18 I = O , then det A is equal to _______.

Step-by-step solution

Given A = 4 - 2 α β   ⇒ det A = 4 β + 2 α Characteristic equation of the matrix is 4 - λ - 2 α β - λ = 0 ⇒ 4 β + λ 2 - β + 4 λ + 2 α = 0 ⇒ λ 2 - β + 4 λ + 2 α + 4 β = 0 Comparing with given equation A 2 + γ A + 18 I = 0 , we get ⇒ γ = - β + 4   &   2 α + 4 β = 18 Hence det A = 4 β + 2 α = 18