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# If A,B are square matrices of same order and B is a skew-symmetric matrix,show that A'BA is skew symmetric.

Toolbox:
• A square matrix A=[a$_{ij}$] is said to be symmetric if A'=A that is $[a_{ij}]=[a_{ji}]$ for all possible value of i and j.
• A square matrix A=[a$_{ij}$] is said to be skew symmetric if A'=-A that is $[a_{ij}]= -[a_{ji}]$ for all possible value of i and j.
Given:
B is a skew symmetric matrix.
$\Rightarrow B=-(B')$
A,B$\rightarrow$ Square matrix
$A'BA\rightarrow$ skew symmetric matrix
$\Rightarrow (A'BA)'=-(A'BA)$
Consider the LHS:-
$(A'BA)'$
$\Rightarrow (BA)'(A')'$ [From the property we have $(AB)^1=B^1A^1$]
$\Rightarrow (BA)'A$
$\Rightarrow A'B'.A$
$\Rightarrow A'(-B).A$ [Replacing $B'=-B$]
$\Rightarrow -A'B.A$
Hence $(A'BA)'=-(A'BA)$
Hence A'BA is skew symmetric matrix