Discussion Forum

Let M and N be two 3x3 non-singular skew-symmetric matrices such that MN = NM. If P^T denotes the transpose of P, then  $M^{2}N^{2}(M^{T}N)^{-1}(MN^{-1})^{T}$ is  equal to 

Answer:

Explanation:

Given    $M^{T}=-M,N^{T}=-N$

 and MN=NM

 $\therefore$    $M^{2}N^{2}(M^{T}N)^{-1}(MN^{-1})^{T}$

$= M^{2}N^{2}N^{-1}(M^{T})^{-1}(N{-1})^{T}.M^{T}$

=$M^{2}N(NN^{-1})(-M)^{-1}(N^{T})^{-1}(-M)$

=$M^{2}NI(-M^{-1})(-N)^{-1}(-M)$

=$-M.(MN)M^{-1}N^{-1}M$

=$-MN(NM^{-1})N^{-1}M$

 =$-M(NN^{-1})M=-M^{2}$

Note Here, non-singular word should not be used, since there is no non-singular 3 x 3 skew-symmetric matrix