JEE Advanced · 2013 · 02 Jun 2013 · Paper 1 · Matrices

For (A): $(N^{T}MN{)}^T=N^T{M}^{T}N$. If $M$ is symmetric, $M^T=M$, so $N^{T}MN$ is symmetric. If $M$ is skew-symmetric, $M^T=-M$, so $N^{T}MN$ is skew-symmetric. Thus (A) is correct. For (B): If $M$ and $N$ are symmetric, $(MN-NM{)}^T=N^T{M}^T-M^T{N}^T=NM-MN=-(MN-NM)$, so it is skew-symmetric. Thus (B) is correct. For (C): $(MN{)}^T=N^T{M}^T=NM$, which equals $MN$ only if $M$ and $N$ commute. This is not true for all symmetric matrices. Thus (C) is NOT correct. For (D): The correct identity is $(\text{adj }M)(\text{adj }N)=\text{adj}(NM)$, not $\text{adj}(MN)$. Thus (D) is NOT correct.