Question

The trace of an n × n-matrix A, denoted tr(A), is defined as the sum of its diagonal entries, i.e., tr(A) a11 a2+ +ann (a) Prove that tr(AB)- tr(BA). (b) Show that if A is similar to B, then tr(A) = tr(B).

Answer 1

Let A be any n times n matrix and trace of A, denoted as 'tr (A)' and is defined as tr A = a_11 + a_22 + Ellipsis + a_nn = sigma^h_i = 1 a_ii = sum of main diagonal elements of A we have to show tr (AB) = tr (BA) let C = AB, then elements of C, say c_ij = sigma_h_k = 1 a_ik b_kj (i) Therefore tr (C) = sigma^n_i = 1 c_ii = sigma^n_i = 1 sigma^n_k = 1 a_ik b_ki Because c_ii = sigma^n_k = 1 a_ik b_ki similarly let D = BA, then d_ii = sigma^n_k = 1 b_ik a_ki Therefore tr (D) = sigma^n_i = 1 d_ii = sigma^n_i = 1 sigma^n_k = 1 b_ik a_ki \r\n tr (D) = sigma^n_i = 1 sigma^n_k = 1 a_ki b_ik, Because b_ik s a_ik are number so bu commutativity property b_ik a_ki = a_ki b_ik = sigma^n_k = 1