Let A and B be two non-singular skew-symmetri | vedclass

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(C) Given that $A$ and $B$ are skew-symmetric matrices, we have $A^{	op} = -A$ and $B^{	op} = -B$. Given $AB = BA$. We need to evaluate the expressi

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$-A^{2}$

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(C) Given that $A$ and $B$ are skew-symmetric matrices, we have $A^{	op} = -A$ and $B^{	op} = -B$. Given $AB = BA$. We need to evaluate the expression $E = A^{2} B^{2} (A^{	op} B)^{-1} (A B^{-1})^{	op}$. Substituting $A^{	op} = -A$, we get $E = A^{2} B^{2} (-AB)^{-1} (A B^{-1})^{	op}$. Using the property $(XY)^{-1} = Y^{-1} X^{-1}$ and $(XY)^{	op} = Y^{	op} X^{	op}$, we have: $E = A^{2} B^{2} (-B^{-1} A^{-1}) ((B^{-1})^{	op} A^{	op})$. Since $(B^{-1})^{	op} = (B^{	op})^{-1} = (-B)^{-1} = -B^{-1}$, we substitute this: $E = A^{2} B^{2} (-B^{-1} A^{-1}) (-B^{-1} (-A))$. $E = A^{2} B^{2} (-B^{-1} A^{-1}) (B^{-1} A)$. Since $AB = BA$, we also have $A^{-1} B = B A^{-1}$ and $A B^{-1} = B^{-1} A$. $E = A^{2} B^{2} (-B^{-1} A^{-1} B^{-1} A)$. $E = -A^{2} B^{2} B^{-1} A^{-1} B^{-1} A$. $E = -A^{2} B (B B^{-1}) A^{-1} B^{-1} A$. $E = -A^{2} B (I) A^{-1} B^{-1} A$. $E = -A^{2} (B A^{-1}) B^{-1} A$. Since $B A^{-1} = A^{-1} B$, we have: $E = -A^{2} A^{-1} B B^{-1} A$. $E = -A (I) (I) A = -A^{2}$.

Let $A$ and $B$ be two non-singular skew-symmetric matrices such that $AB = BA$. Then $A^{2} B^{2} (A^{\top} B)^{-1} (A B^{-1})^{\top}$ is equal to

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