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

Step-by-step solution

Given, M^T=-M, N^T=-N and M N=N M aligned & M^2 N^2 (M^T N )⁻¹ (M N⁻¹ )^T & =M^2 N^2 N⁻¹ (M^T )⁻¹ (N⁻¹ )^T M^T & =M^2 N (N N⁻¹ )(-M)⁻¹ (N^T )⁻¹(-M) & =M^2 N (-M⁻¹ )(-N)⁻¹(-M) & =-M^2 N M⁻¹ N⁻¹ M & =-M (M N) M⁻¹ N⁻¹ M & =-M(N M) M⁻¹ N⁻¹ M & =-M N (N M⁻¹ ) N⁻¹ M & =-M (N N⁻¹ ) M=-M^2 aligned Note Here, non-singular word should not be used, since there is no non-singular 3 3 skew-symmetric matrix.