Answer to Question #131499 in Linear Algebra for Axwell

Question

Suppose A and B are 3 × 3 matrices. Prove that
(i) (A + B)^T = B^T + A^T

(ii) det (AB)^T = det (A). det (B)

Answer 1

Solution.

(i) Let A = (a_ij )_n×n , A^T = (a'_ij )_n×n , B = (b_ij )_n×n , B^T = (b'_ij )_n×n , (A+B) = (c_ij ) and (A+B) ^T = (d_ij ). Then d_ij = c_ji = a_ji +b_ji = (a'_ij +b'_ij ). Therefore, (A + B) ^T = B^T +A^T

(ii) det(AB)^T = detA^T. detB^T

Hence, detA^T= detA, detB^T= detB

Therefore, det(AB)^T= det(A). det(B)