A) \[{{N}^{T}}MN\]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) none of these
Correct Answer: C
Solution :
[a] \[\left( N'MN \right)'=\left( MN \right)'N=N'M'N=N'MN\] \[\operatorname{or}\,-N'MN\] According as M is symmetric or skew symmetric \[\therefore \]correct [b] \[(MN-NM)'=(MN)'-(NM)'\] \[=N'M'-M'N'\] \[=NM-MN=-(MN-NM)\] \[\therefore \]It is skew symmetric \[\therefore \]Correct. [c] \[(MN)'=N'M'=NM\ne MN\] \[\therefore \]incorrect
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