For 3 × 3 matrices M and N, which of the following statement(s) is (are) not correct?
I. $N^T MN$ is symmetric skew-symmetric, according as M is symmetric or skew-symmetric.
II. $MN- NM$ is skew symmetric for all symmetric matrices M and N.
III. MN is symmetric for all symmetric matrices M and N.
IV. (adj M) (adj N) = adj (MN) for all invertible matrices M & N.
Answer & explanation
Correct answer: option 3
Clearly, $(N^TMN)^T = N^TM^T (N^T)^T = N^TM^TN$
$⇒(N^TMN)^T=\left\{\begin{matrix}N^T MN,\, if\, M^T = M\, i.e.\, M\, is\, symmetric\\-N^T MN,\, if\, M^T =-M\, i.e.\, M\, is\, skew-symmetric\end{matrix}\right.$
So, statement I is correct.
$(MN-NM)^T=(MN)^T -(NM)^T$
$=N^TM^T – M^TN^T$
$=NM-MN$ [∵ N & M are symmetric matrices]
$=-(MN-NM)$
∴ MN - NM is skew-symmetric matrix.
If M and N are symmetric matrices, then
$(MN)^T =N^TM^T = NM≠ MN$
So, MN is not a symmetric matrix.
Hence, statement III is not correct.
We know that adj (MN) = (adj N) (adj M). So, statement IV is not correct.