Let T be an n × n matrix with some Jordan canonical form J. Recall that the sizes of the Jordan blocks of J corresponding to an eigenvalue λ of T are completely determined by the sequence of integers dim(ker(T − λI) ⁿ).

(a) Prove that if T, S are similar (i.e., there is an invertible matrix U such that S = UT U⁻¹), then for all n ∈ N, dim(ker(T − λI)ⁿ) = dim(ker(S − λI)ⁿ).

(b) Conclude that if S, T are similar, then they have the same Jordan canonical form J.

(c) Recall that rank(A) = rank(A^t) for any matrix A. Use this and the rank-nullity theorem to prove that T and T^t have the same Jordan canonical form.