Question 1006564
First part of the statement


Given:
B is square


since B is square, this means B*B^T is defined


we want to prove
(B*B^T)^T = B*B^T
is true


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(B*B^T)^T = B*B^T
(B^T)^T*B^T = B*B^T ... use <a href = "https://en.wikipedia.org/wiki/Transpose">property 3</a>
B*B^T = B*B^T ...  ... use <a href = "https://en.wikipedia.org/wiki/Transpose">property 1</a>; we end up with a true equation


So the first part of the statement is true.


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Now onto the second part of the statement


B is square ----> B+B^T is defined


we want to prove B+B^T is symmetric, so we want to show that 
(B+B^T)^T = B+B^T
is true


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(B+B^T)^T = B+B^T
B^T+(B^T)^T = B+B^T ... use <a href = "https://en.wikipedia.org/wiki/Transpose">property 2</a>
B^T+B = B+B^T  ... use <a href = "https://en.wikipedia.org/wiki/Transpose">property 1</a>
B+B^T = B+B^T ... commutative property of matrix addition; true equation


The last true equation shows that (B+B^T)^T = B+B^T is true as well. So the second part of the statement is true.

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Both parts of the statement have been proven true. Overall, the statement is true.