Question

1) Let A,B and C are three matrices of the
same order such that any two are symmetric
and third one is skew symmetric if X=ABC+
CBA and Y= ABC-CBA then (XY) transpose is​

Answer 1

Matrix

Formula:

• If $A$ be a symmetric matrix, then $A^{T}=A$.

• If $A$ be a skew-symmetric matrix, then $A^{T}=-A$.

• $(A+B)^{T}=A^{T}+B^{T}$

• $(kA)^{T}=kA^{T}$, where $k$ is a scalar.

• $(ABC)^{T}=C^{T}B^{T}A^{T}$

Solution:

Let us take $A,\:B$ being symmetric matrices and $C$ being a non-symmetric matrix.

Then $A^{T}=A,\:B^{T}=B,\:C^{T}=-C$

Given, $X=ABC+CBA,\:Y=ABC-CBA$

$\therefore (XY)^{T}$

$=Y^{T}X^{T}$

$=(ABC-CBA)^{T}\:(ABC+CBA)^{T}$

$=\{(ABC)^{T}-(CBA)^{T}\}\:\{(ABC)^{T}+(CBA)^{T}\}$

$=(C^{T}B^{T}A^{T}-A^{T}B^{T}C^{T})\:(C^{T}B^{T}A^{T}+A^{T}B^{T}C^{T})$

$=(-CBA+ABC)\:(-CBA-ABC)$

$=-(ABC-CBA)\:(ABC+CBA)$

$=-YX$

$\Rightarrow \boxed{(XY)^{T}=-YX}$

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