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Show that A′A and AA′ are both symmetric matrices for any matrix A.

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Question

Show that A′A and AA′ are both symmetric matrices for any matrix A.

Sum
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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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Chapter 3: Matrices - Exercise [Page 56]

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NCERT Exemplar Mathematics Exemplar [English] Class 12
Chapter 3 Matrices
Exercise | Q 29 | Page 56

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