SOLUTION: A square matrix A is called symmetric if A^t = A. If B is a square matrix, then B.B^t and B + B^t are symmetric is this statement true or false and why?

Algebra ->  College  -> Linear Algebra -> SOLUTION: A square matrix A is called symmetric if A^t = A. If B is a square matrix, then B.B^t and B + B^t are symmetric is this statement true or false and why?       Log On


   

Question 1006564: A square matrix A is called symmetric if A^t = A. If B is a square matrix, then B.B^t and B + B^t are 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!
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 property 3
B*B^T = B*B^T ... ... use property 1; 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

------------------------------

(B+B^T)^T = B+B^T
B^T+(B^T)^T = B+B^T ... use property 2
B^T+B = B+B^T ... use property 1
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.