Question

Prove that if A and B are matrices such that AB and BA both exist, then AB and BA have the same sum of diagonal elements, i.E., tr(AB) = tr(BA). Extend the result to show that tr(ABC) = tr(BCA), provided the matrices are conformable for multiplication (15 points).

Answer 1

q.e.d.

Explanation:

since AB and BA both exist, then A and B both have same size. since AB has main diagonal entry product of i-th row of A and i-th colom of B while BA has main diag entry product of i-th row of B and i-th colom of A, then tr(AB) = tr(BA) by using some algebraic manipulations. substitute B := BC to get tr(ABC) = tr(BCA)