Given the matrix A = [5 6 2] [0 -1 -8] [1 0 -2] (a) Find the characteristic poly

Question

Given the matrix
A =
[5 6 2]
[0 -1 -8]
[1 0 -2]

(a) Find the characteristic polynomial, p(?).
(b) Find the eigenvalues of A. (You may use a tool to factor the polynomial.)
(c) Based on these eigenvalues, is A invertible? Explain why or why not.
(d) Find a set of eigenvectors associated with the above eigenvalues. Be sure to note which
eigenvectors associate with which eigenvalues.
(e) How many eigenspaces does A have?
(f) In a smart way, find the eigenvalues of A2.
(g) What are the algebraic and geometric multiplicities of the eigenvalues of matrix A?
(h) Is A diagonalizable? Explain why or why not.

Explanation / Answer

Characteristic polynomial: x^3 - 2x^2 - 15x + 36 Real eigenvalues: {-4, 3, 3} Eigenvector of eigenvalue -4: (-6, 8, 3) Eigenvector of eigenvalue 3: (5, -2, 1) Eigenvector of eigenvalue 3: (5, -2, 1)

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