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For 3 x 3 matrices M and N, which of the following statement(s) is (are) NOT correct ? (A) NTMN is symmetric or skew symmetric, according as M is symmetric or skew symmetric (B) MN - NM is skew symmetric for all symmetric matrices M and N (C) MN is symmetric for all symmetric matrices M and N (D) (adj M) (adj N) = adj(MN) for all invertible matrices M and N |
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Answer» (C) MN is symmetric for all symmetric matrices M and N (D) (adj M) (adj N) = adj(MN) for all invertible matrices M and N (A) (NTMN)T = NTM TN = NTMN if M is symmetric and is – NTMN if M is skew symmetric (B) (MN - NM)T = NTMT - MTNT = NM - MN = –(MN – NM). So, (MN – NM) is skew symmetric (C) (MN)T = NTMT = NM ≠ MN if M and N are symmetric. So, MN is not symmetric (D) (adj. M) (adj. N) = adj(NM) ≠ adj (MN). |
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