Electronics and Communication Engineering - Exam Questions Papers - Discussion

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).