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Question
Show that A′A and AA′ are both symmetric matrices for any matrix A.
Solution
Let P = A'A
⇒ P' = (A'A)'
⇒ P' = A'(A')' .....[(AB') = B'A']
⇒ P' = A'A ......[∵ (A')' = A]
⇒ P' = P
Hence, A'A is a symmetric matrix.
Now, Let Q = AA'
⇒ Q' = (AA')'
⇒ Q' = (A')A' .....[(AB)' = B'A']
⇒ Q' = AA' ......[∵ (A')' = A]
⇒ Q' = Q
Hence, AA' is also a symmetric matrix.
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