Let A = 1 2 - 2 - 5 . Let α , β ∈ ℝ be such that α A 2 + β A = 2 I . Then α + β is equal to

Step-by-step solution

We know that characteristic equation of matrix A is given by, A - λ I = 0 ⇒ 1 - λ 2 - 2 - 5 - λ = 0 ⇒ λ 2 + 4 λ = 1 ⇒ A 2 + 4 A = I Multiplying by 2 both side we get, ⇒ 2 A 2 + 8 A = 2 I             ⋯ 1 Given that α A 2 + β A = 2 I           ⋯ 2 Now on Comparing equation 1   &   2 we get α = 2 ,   β = 8 ∴     α + β = 10