O 5. [8pt] Let A = (a) [1.5pt] (a) Find the characteristic polynomial of A and show that the eigenvalues of A are 1 and -1 and 2. Then find an algebraic multiplicity of each eigenvalue. (b) [1.5pt] Find a basis for E1 = {x e R'[Ax = x}. What is the geometric multiplicity of eigenvalue 1? Explain your answer. (c) [1.5pt] Find a basis for Ez = {x ( R3|Ar = 2x}. what is the geometric multiplicity of eigenvalue 2? Explain your answer. (d) [1.5pt] Find a basis for E_1 = {x ( R3|Ar = -x}. what is the geometric multiplicity of eigenvalue -1? Explain your answer. (e) [2pt] If possible, find an invertible matrix P and a diagonal matrix D such that P-AP = D. You must explain why the chosen matrix P is invertible. If A is not diagonalizable, you must explain why. (To find P is invertible, you do not need to find inverse of P, you can verify it by its determinant value.)