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asked in Linear Algebra by Boss (6k points)  
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+5 votes
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Given expression : M2 N2 (MTN) -1 (MN-1)T

                          =       M2N(-MN)-1  (MN-1)T   [ Since M is a skew symmetric matrix , so MT  = -M]

                   =       - M2N . N-1M-1 .(NT)-1 MT 

    [Since for any two matrices A and B , (AB)-1 = B-1A-1   and 

                                                           (AB)T  = BT AT    and 

                                                           (A-1)T  = (AT)-1     ]

                  =     - M2 N2 N-1 M-1  (-N)-1(-M)

                  =     - MM N M-1 (-N)-1(-M)  [ Since for any matrix A , A A-1 = I where I is identity matrix]

                  =     - M N M M-1 (-N)-1(-M)  [Since given in the question MN = NM]

                        =     - M2                        [Since middle M,M-1 and N,N-1 are eliminated using A A-1 = I]

 

Hence  C) option should be the correct option.

answered by Veteran (68.7k points)  
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$(AB)^T = B^TA^T$
Corrected @mcjoshi ..:)


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