A square matrix N is called nilpotent if there exists some positive integer k such that Nk = 0. Prove that if N is a nilpotent matrix, then the system Nx = 0 has nontrivial solutions.

Nx = λx

Nx = 0, with x≠0

if N is nilpotent matrix, then the system Nx = 0 has non-trivial solutions

Step-by-step explanation:

given that

let N be a square matrix in order of n

note: N is nilpotent matrix with , k ∈ N

let λ be eigenvalue of N and let x be eigenvector corresponding to eigenvalue λ

Nx = λx (x≠0)

N²x =  λNx = λ²x

∴ =  (λ^k)x

= 0, (λ^k)x = , where n is dimensional vector

where x = 0, (λ^k) = 0

λ = 0

therefore, Nx = λx

Nx = 0, with x≠0

note: if N is nilpotent matrix, then the system Nx = 0 has non-trivial solution

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step-by-step explanation: