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  4. Let A and B be two non-singular skew symmetric matrices such that A B=B A, then A2 B2(A T B)-1(A B-1) T is equal to

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Q. Let $A$ and $B$ be two non-singular skew symmetric matrices such that $A B=B A$, then $A^{2} B^{2}\left(A^{ T } B\right)^{-1}\left(A B^{-1}\right)^{ T }$ is equal to

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Solution:

$A B=B A, A^{ T }=-A, B^{ T }=-B$
$\because$ matrices are non singular
$\therefore $ order is even
$A^{2} B^{2}\left(A^{ T } B\right)^{-1}\left(A B^{-1}\right)^{ T }$
$=A^{2} B^{2}(-A B)^{-1}\left(A B^{-1}\right)^{ T }$
$=-A^{2} B^{2} B^{-1} A^{-1}\left(B^{ T }\right)^{-1} A^{ T }$
$=-A^{2} B A^{-1} B^{-1} A$
$=-A^{2} B(B A)^{-1} A$
$=-A^{2} B(A B)^{-1} A$
$=-A^{2} B B^{-1} A^{-1} A=-A^{2}$