Determining Symmetry of Matrix F: ABA

In summary, the question is asking whether the matrix F=ABA is symmetric or nonsymmetric if A and B are symmetric n x n matrices. The attempted solution involves using the property (AB)^T=B^T A^T and distributing the terms to show that F is symmetric. It is uncertain if this method is valid.
  • #1
mvpshaq32
28
0

Homework Statement



Let A and B be symmetric n x n matrices. Determine whether the given matrix must be symmetric or could be nonsymmetric.

F=ABA

Homework Equations



(AB)^T=B^T A^t


The Attempt at a Solution



So if it's symmetric, that means (ABA)^T=ABA. I decided to make A one term and BA another term and distributed it to be (BA)^T (A)^T=A^T B^T A^T=ABA, making it symmetric.
But I'm not sure if I can evaluate it like this.
 
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  • #2
That looks perfectly ok to me.
 
  • #3
Dick said:
That looks perfectly ok to me.

Thanks very much!
 

FAQ: Determining Symmetry of Matrix F: ABA

What is a matrix?

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It is commonly used in mathematics and computer science to represent data and perform operations on that data.

What is symmetry in a matrix?

A matrix is said to be symmetric if it is equal to its own transpose. This means that if you were to reflect the matrix over its main diagonal, the values on either side would be the same.

How do you determine the symmetry of a matrix?

To determine the symmetry of a matrix, you need to compare the matrix to its transpose. If the values in the matrix and its transpose are the same, then the matrix is symmetric. If they are not the same, then the matrix is not symmetric.

What is the significance of a symmetric matrix?

Symmetric matrices have many important properties and applications in mathematics and science. For example, they are commonly used in linear algebra to represent quadratic forms and in mechanics to represent physical systems with symmetric properties.

Can a matrix be both symmetric and non-symmetric?

No, a matrix cannot be both symmetric and non-symmetric. If a matrix is symmetric, it means that it is equal to its own transpose. Therefore, if a matrix is not equal to its transpose, it cannot be symmetric.

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