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  4. A skew-symmetric matrix M satisfies the relation M+I=0 , where I is the unit matrix. Then, MM' is equal to

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Q. A skew-symmetric matrix $ M $ satisfies the relation $ M+I=0 $ , where $ I $ is the unit matrix. Then, $ MM' $ is equal to

Jharkhand CECEJharkhand CECE 2012

Solution:

Since, $ M $ is skew-symmetric matrix
$ M'=-M $ Also, $ {{M}^{2}}+I=0 $
$ \Rightarrow $ $ {{M}^{2}}=-I+0=-I $
$ \Rightarrow $ $ M\cdot M=-I $
$ \Rightarrow $ $ M\,\,M\,\,M'=IM'=I(-M')=IM=M $
$ \Rightarrow $ $ {{M}^{-1}}MMM'={{M}^{-1}}M $
$ \Rightarrow $ $ IMM'=I $
$ \Rightarrow $ $ MM'=I $