SOLUTION: If A and B are symmetric matrices and if AB = BA, then AB is symmetric is this statement true or false and why?

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Question 1006565: If A and B are symmetric matrices and if AB = BA, then AB is symmetric

is this statement true or false and why?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
A matrix is symmetric if and only if it is equal to its transpose, ie
X = X^T

Given:
A = A^T (since matrix A is symmetric)
B = B^T (matrix B is symmetric)
AB = BA

We want to prove:
AB is symmetric
ie, AB = (AB)^T


AB = BA
AB = B^T*A^T ... use the given info above
AB = (AB)^T ... use property 3

So the claim has been proven true. AB is indeed symmetric.