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Question:

For 3x3 matrices M and N, which of the following statement(s) is (are) NOT correct?

MN-NM is skew-symmetric for all symmetric matrices M and N

NTMN is symmetric or skew-symmetric, according to as M is symmetric or skew-symmetric

MN is symmetric for all symmetric matrices M and N

(adjM)(adjN)=adj(MN) for all invertible matrices M and N

Solution:

(A) (NTMN)T = NTMTN = NTMN, if M is symmetric
(NTMN)T = NTMTN = -NTMN, if M is skew symmetric
(B) (MN-NM)T = NTMT - MTNT = NM - MN = -(MN - NM). So, (MN-NM) is skew symmetric