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(A) Given the equation: $(AB+BA)^{T}+(AB-BA)^{T}=2BA$ Using the property $(X+Y)^T = X^T + Y^T$,we get: $(AB)^T + (BA)^T + (AB)^T - (BA)^T = 2BA$ $2(AB

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$A$ and $B$ are both symmetric matrices but not skew-symmetric matrices

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(A) Given the equation: $(AB+BA)^{T}+(AB-BA)^{T}=2BA$ Using the property $(X+Y)^T = X^T + Y^T$,we get: $(AB)^T + (BA)^T + (AB)^T - (BA)^T = 2BA$ $2(AB)^T = 2BA$ $(AB)^T = BA$ $B^T A^T = BA$ If $A$ and $B$ are symmetric,then $A^T = A$ and $B^T = B$. Substituting these,we get $BA = BA$,which is true. If $A$ and $B$ are skew-symmetric,then $A^T = -A$ and $B^T = -B$. Substituting these,we get $(-B)(-A) = BA$,which is $BA = BA$,which is also true. Thus,the condition holds if $A$ and $B$ are both symmetric or both skew-symmetric. Since the options provided are limited,the most appropriate conclusion based on the derivation is that the property holds for both cases.

If $A$ and $B$ are two square matrices of the same order and $(AB+BA)^{T}+(AB-BA)^{T}=2BA$,then:

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