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  4. A, B, C are three matrices of the same order such that any two are symmetric and the 3 text rd one is skew symmetric. If X=A B C+C B A and Y=A B C-C B A, then (X Y)T is

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Q. $A, B, C$ are three matrices of the same order such that any two are symmetric and the $3^{\text {rd }}$ one is skew symmetric. If $X=A B C+C B A$ and $Y=A B C-C B A,$ then $(X Y)^{T}$ is

Matrices

Solution:

$(X Y)^{T}=Y^{T} X^{T}$
$Y^{T}=(A B C-C B A)^{T}$
$=C^{T} B^{T} A^{T}-A^{T} B^{T} C^{T}$
$=-C B A+A B C=Y$
$X^{T}=(A B C+C B A)^{T}$
$=C^{T} B^{T} A^{T}+A^{T} B^{T} C^{T}$
$=-C B A-A B C=-X$