If A and B are symmetric matrices of the same | vedclass

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(C) Given that $A$ and $B$ are symmetric matrices,we have $A^{T} = A$ and $B^{T} = B$.We are given $X = AB + BA$ and $Y = AB - BA$.We need to find $(X

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$-YX$

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(C) Given that $A$ and $B$ are symmetric matrices,we have $A^{T} = A$ and $B^{T} = B$.We are given $X = AB + BA$ and $Y = AB - BA$.We need to find $(XY)^{T}$.Using the property of transpose $(XY)^{T} = Y^{T} X^{T}$.First,let us find $X^{T}$ and $Y^{T}$:$X^{T} = (AB + BA)^{T} = (AB)^{T} + (BA)^{T} = B^{T}A^{T} + A^{T}B^{T} = BA + AB = X$.$Y^{T} = (AB - BA)^{T} = (AB)^{T} - (BA)^{T} = B^{T}A^{T} - A^{T}B^{T} = BA - AB = -(AB - BA) = -Y$.Now,$(XY)^{T} = Y^{T} X^{T}$.Substituting the values of $Y^{T}$ and $X^{T}$,we get:$(XY)^{T} = (-Y)(X) = -YX$.

If $A$ and $B$ are symmetric matrices of the same order such that $AB+BA=X$ and $AB-BA=Y$,then $(XY)^{T}=$

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