Consider the following two statements with respect to the matrices A_mxn, B_nxm, C_nxn and D_nxn: Statement 1: tr(AB) = tr(BA) Statement 2: tr(CD) = tr(DC) where tr() represents the trace of a matrix. Which one of the following holds? A. Statement 1 is correct and Statement 2 is wrong. B. Statement 1 is wrong and Statement 2 is correct. C. Both Statement 1 and Statement 2 are correct. D. Both Statement 1 and Statement 2 are wrong.

GATE 2022 · Engineering Mathematics · Matrix · medium

Answer: Both Statement 1 and Statement 2 are correct. Answer: C

1. Prove tr(AB) = tr(BA) for rectangular A (mxn), B (nxm): = sum_{k=1}^{n} sum_{i=1}^{m} B_{ki} A_{ik} = sum_{k=1}^{n} (BA)_{kk} = tr(BA). Statement 1 is TRUE.
2. Statement 2 follows as a special case: With m = n in the above proof, square matrices C, D also satisfy tr(CD) = tr(DC). Statement 2 is TRUE.