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Find eigenvalue of A^3 - 2A + 4I given eigenvalue of A?

Update:

A is a 2X2 matrix. It has an eigenvalue = 2 with associated eigenvector

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Find an eigenvalue and the associated eigenvector of B = A^3 - 2A + 4I

1 Answer

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  • ?
    Lv 7
    4 years ago

    If λ,x are an eigenpair of A then Ax=λx

    Multiply by A repeatedly and use Ax=λx to get Aⁿx=λⁿx

    ∴ (A³−2A+4)x = A³x−2Ax+4x = (λ³−2λ+4)x

    This means λ³−2λ+4,x are an eigenpair for B

    For x=<1,-1>, λ=2 you get for B <1,-1> and 8-4+4=8

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