If xx, yy, and zz are the eigenvalues of the matrix AA, what are the eigenvalues of the matrices AnA^n and A+cIdA + c{\rm Id}, respectively.

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Use the definition of eigenvalue. You know that Av=xvAv = xv for some vector vv, so Anv=An−1xv=An−2x2v=…=xnvA^n v = A^{n-1} xv = A^{n-2}x^2v = \ldots = x^nv etc. figure out the rest and the answer to the second part.

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2 days ago

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