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

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Q.

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

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a

N^TMN 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

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

answer is C, D.

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Detailed Solution

a) ${(N^{1} MN)}^{1} = (MN)^{1} N = N^{1} M^{1} N = N^{1} MN or - N^{1} MN$
according as M is symmetric or skew symmetric.
b) $(MN - NM)^{1} = (MN)^{1} - (NM)^{1} = N^{1} M^{1} - M^{1} N^{1} = NM - MN$
skew symmetric Correct
c) $(MN)^{1} = N^{1} M^{1} = NM \neq MN ∴$ Incorrect
d) (adjM)(adjN)=(adjMN) $∴$ Incorrect

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