If A = 1 5 λ 10 , A - 1 = α A + β I and α + β = - 2 , then 4 α 2 + β 2 + λ 2 is equal to :

Step-by-step solution

Given, A = 1 5 λ 10 ,   A - 1 = α A + β I and α + β = - 2 , Now solving by using characteristic equation, we get | A - k | = 0 ⇒ 1 - k 5 λ 10 - k = 0 ⇒ k 2 - 11 k + 10 - 5 λ = 0 ⇒ A 2 - 11 A + ( 10 - 5 λ ) I = 0 by putting k = A ⇒    A - 1 = 1 10 - 5 λ ( - A + 11 I ) Now on comparing with A - 1 = α A + β I we get, ⇒ α = - 1 10 - 5 λ ,   β = 11 10 - 5 λ And given, α + β = - 2 ⇒