1. Let T : Mn×n(F) ? Mn×n(F) be the transposition map, T(A) =At. Compute the characteristic polynomial of T. You may wish to usethe basis of Mn×n(F) consisting of the matrices eij + eji, eij ?ejiand eii.
2. Let A = (a b c d) (2 by 2 matrix) and let T :M2×2(F) ? M2×2(F) be de?ned asT (B) = AB. Represent T as a 4×4matrix using the ordered basis {e11,e21,e12,e22}, and use thismatrix to prove that the characteristic polynomial of T is thesquare of the characteristic polynomial of A.
