The trace of an (n x 71) matrix A is dened to be tr(A) : i a,,. 21 That is, the trace of a square matrix is simply the sum of its diagonal elements. The trace of a scalar is the scalar itself. a. If A and B are square matrices of the same dimensions, Show that tr(A + B): tr(A) + tr(B). b. If A is a square matrix, show that tr(AT) : tr(A). c. If A is (m x n) and B is (n x m), Show that tr(AB) : tr(BA). d. If a: is an (n x 1) column vector, Show that xTx : tr(xxT). Show this by (i) direct multiplication, (ii) using the result in part(c) and the fact that the trace of a scalar is the scalar itself