Let A and B be two symmetric matrices. Prove that AB = BA if and only

1.	Let A and B be two symmetric matrices. Prove that AB = BA if and only if AB is a symmetric matrix.
Answer» Let A and B be two symmetric matrices ⇒ A^T = A and B^T = B ... (1) Given that AB = BA (2) To prove AB is symmetric: Now (AB)^T = B^TA^T = BA (from(1)) But (AB)^T = AB by ... (2) ⇒ AB is symmetric. Conversely let AB be a symmetric matrix. ⇒ (AB)^T = AB i.e. B^T A^T = AB i.e. BA = AB (from (1)) ⇒ AB is symmetric

Let A and B be two symmetric matrices. Prove that AB = BA if and only if AB is a symmetric matrix.

Answer»

Let A and B be two symmetric matrices

⇒ A^T = A and B^T = B ... (1)

Given that AB = BA (2)

To prove AB is symmetric:

Now (AB)^T = B^TA^T = BA

(from(1)) But (AB)^T = AB by ... (2)

⇒ AB is symmetric.

Conversely let AB be a symmetric matrix.

⇒ (AB)^T = AB

i.e. B^T A^T = AB

i.e. BA = AB (from (1))

⇒ AB is symmetric

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Let A and B be two symmetric matrices. Prove that AB = BA if and only if AB is a symmetric matrix.

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