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

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

  1. NT MN is symmetric or skew-symmetric, according to as M is symmetric or skew-symmetric
  2. ​MN - NM is skew symmetric for all symmetric matrices M and N
  3. MN is symmetric for all symmetric matrices M and N
  4. adj(M) adj(N) = adj (MN) for all invertible matrices M and N

Solution

N T MN T = N T M T N  is symmetric if M is symmetric and skew-symmetric if M is skew-symmetric.
MN - NM T = MN T - NM T
                          = NM - MN
                          = - MN - NM
skew symmetric
MN T = N T M T
            =  NM
             ≠  MN  hence not correct
Standard result is
adj MN = adj N adj M
≠ adj M adj N

Asked in: JEE Advanced 2013 (Paper 1)

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