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Q5. Show that the matrix B′AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.

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Let be a A is symmetric matrix , then $A'=A$

Consider, $(B'AB)' ={B'(AB)}'$

$={(AB)}'(B')'$

$= B'A'(B)$

$= B'(A'B)$

Replace $A'$ by $A$

$=B'(AB)$

i.e. $(B'AB)'$ $=B'(AB)$

Thus, if A is symmetric matrix than $B'(AB)$ is a symmetric matrix.

Now, let A be a skew symmetric matrix, then $A'=-A$ .

$(B'AB)' ={B'(AB)}'$

$={(AB)}'(B')'$

$= B'A'(B)$

$= B'(A'B)$

Replace $A'$ by - $A$ ,

$=B'(-AB)$

$= - B'AB$

i.e. $(B'AB)'$ $= - B'AB$ .

Thus, if A is skew-symmetric matrix then $- B'AB$ is a skew symmetric matrix.

Hence, the matrix B′AB is symmetric or skew-symmetric according to as A is symmetric or skew-symmetric.

Posted by

seema garhwal

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