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

Question

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 (A) symmetric (B) skew symmetric (C) I-X Y (D) -Y X

Tardigrade Admin · Accepted Answer

(X Y)T=YT XT YT=(A B C-C B A)T =CT BT AT-AT BT CT =-C B A+A B C=Y XT=(A B C+C B A)T =CT BT AT+AT BT CT =-C B A-A B C=-X

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

symmetric

skew symmetric

$I - X Y$

$- Y X$

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

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