Let A, B and C be matrices such that the following operations are defined. Let A and tr(A) denote the transpose and trace of A respectively. Let I and O denote identity and zero matrices respectively. For each of the following statements, determine whether it is True or False. If it is True, prove it. If it is False, give a counter example to disprove it. Four or five randomly selected statements will be graded for 20 points. (1) If AB= O, then A = O or B = 0. (2) If A²= I, then A= I or A=-1. (3) AB=BA (4) If AB AC, then B = C. = (5) A(BC) (6) (AB)' = B' A' (7) A+ A¹ is symmetric. (8) A-A' is skew-symmetric. (9) If A and B are symmetric, then so is AB. (10) If A = (ai)nxn is skew-symmetric, then a = 0 for all 1 ≤ i ≤n. (11) tr(A + B)=tr(A) + tr(B) (12) tr(aA) = a"tr(A), where a R and A has size nx n. (AB)C (13) tr(AB) = tr(A)tr(B) (14) tr(AB)= tr(BA) (15) tr(AA) = (tr(A))2. (16) There exist square matrices A and B such that AB-BA = I.