Construct the following examples (You must justify your examples): (a) Two matrices with the same characteristic polynomials x 2 (x 1) but not...

Construct the following examples (You must justify your examples):

(a) Two matrices with the same characteristic polynomials x2(x − 1) but not similar.

(b) A linear endomorphism T : R3 → R3 such that the characteristic polynomial of T is (x − 1)(x + 1)2 and T2 =/ 1.

(c) Two 3 × 3 matrices A and B such that both A and B arediagonalizable but A + B is not diagonalizable.

(d) Two 3 × 3 matrices A and B such that both A and B arediagonalizable but AB is not diagonalizable.

(e) Four 2 × 2 matrices A1, A2, B1, B2 such that A1 ∼ B1 and A2 ∼ B2 but A1 + A2 ∼/ B1 + B2.