For any n×mn×m matrix A=(aij)A=(aij) in Matn,m(R)Matn,m(R), define its transpose AtAt be the m×nm×n matrix B=(bij)B=(bij) so that bij=ajibij=aji.
(a) Show that the map
T:Matn,m(R)→Matm,n(R);A↦AtT:Matn,m(R)→Matm,n(R);A↦At
is an injective and surjective linear map.
(b) Let A∈Matn,m(R)A∈Matn,m(R) and B∈Matm,p(R)B∈Matm,p(R) be an n×mn×m and a m×pm×p matrix, respectively. Show
(AB)t=BtAt.(AB)t=BtAt.
(c) Show for any A∈Matn,m(R)A∈Matn,m(R) that
(At)t=A.(At)t=A.
(d) Show that if A∈Matn,n(R)A∈Matn,n(R) is invertible, then AtAt is also invertible and
(At)−1=(A−1)t