4. Let A be an invertible matrix having A1, ..., Xn as its eigenvalues. a) Find all the eigenvalues of A-1. b) Let p(A) = akAk + ak-1Ak-1 +...+ aoIn (a polynomial of the matrix A) where each coefficient a; is a real number. Show that p(di) = ak^ + ak-1X; of A;) is an eigenvalue of p(A) for i = 1, 2, ..., n. k-1 +...+ ao (a polynomial c) Suppose that A is diagonalizable. Show that the p(A) in (2) is also diagonalizable.

Q: For which value of k does the matrix have one real eigenvalue of multiplicity 2? - -5 -4 k 3

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Q: (e)  Show that the matrix is diagonalizable and find an invertible matrix P and a diagonal matrix D…

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Q: Consider the upper-triangular matrix A = -2 -8 -10 0 0 -20 004

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A: Use simple matrix theory of eigenvalues to come up with this example

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A: For example: Let            A=5-10-15000-2

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A: So the eigen value of this matrix given by.