Aⁿ = ? Every n x n matrix satisfy its characteristic equation |A - λI| = 0 λ -> eigen vector A - λI = |A - λI| = = 0 ∴ 1 =, ∴ f(A) = Aⁿ = β₀I + β₁A Replace A by 1, I by 1 f(λ) = λⁿ = β₀ + β₁λ Differentiate w.r.t. λ nλ^{n - 1} = β₁ β₁ = ∴ β₀ = λⁿ - β x λ β₁ = β₀ = - = [1 - n] ∴ Aⁿ = [1 - n]+ . 2n ∴ Aⁿ = = ∴ A⁵⁰ = |A⁵⁰| = (1 - 50²).