JEE Main · 2023 · 25 Jan 2023 · Shift 2 · Matrices

Powers behave predictably under transposition. Since $A^T=A$, every power $A^k$ is symmetric; since $B^T=-B$, the even power $B^{26}$ is symmetric, whereas the odd power $C^{13}$ satisfies $(C^{13}{)}^T=(-C{)}^{13}=-C^{13}$ and is skew-symmetric. For the first statement put $P=A^{13}{B}^{26}-B^{26}{A}^{13}$; then $P^T=B^{26}{A}^{13}-A^{13}{B}^{26}=-P$, so $P$ is skew-symmetric, not symmetric. For the second put $Q=A^{26}{C}^{13}-C^{13}{A}^{26}$; then $Q^T=(-C^{13})A^{26}-A^{26}(-C^{13})=A^{26}{C}^{13}-C^{13}{A}^{26}=Q$, so $Q$ is symmetric. Only the second statement is true.