# Proving Symmetry of A + A(Transpose) for Any Square Matrix A

• HKP
In summary, to show that A + A(transpose) is symmetric for any square matrix A, we must start with the definitions of AT and symmetric matrix. The definition of a symmetric matrix is that A = AT, not A = A(transpose) as previously stated. Therefore, A + A(transpose) = A + A = A (since A = AT), proving that A + A(transpose) is symmetric for any square matrix A.
HKP

## Homework Statement

Show that A + A(transpose) is symmetric for any square matrix A.

## Homework Equations

A symmetric matrice is equal to A=A(transpose)

## The Attempt at a Solution

I said A + A(transpose)
= A(transpose) + A(transpose)
= (A + A) (transpose)
= A + A

I don't know what else to do

It looks like you're assuming that A is symmetric right at the start, which isn't information that you're given.

The definition you show for a symmetric matrix isn't right - I think you want this:
A symmetric matrix A is such that A = AT. (A matrix can't be "equal" to an equation.)

## 1. How do you prove symmetry of A + A(Transpose) for any square matrix A?

To prove symmetry of A + A(Transpose), we need to show that (A + A(Transpose))(Transpose) = A + A(Transpose). This can be done by expanding the left side of the equation and showing that it is equal to the right side. This proves that the matrix is symmetric.

## 2. Why is it important to prove symmetry of a matrix?

Proving symmetry of a matrix is important because it provides information about the matrix's properties and structure. Symmetric matrices have many useful properties and are often easier to work with in calculations and applications.

## 3. Can you provide an example of a symmetric matrix?

Yes, an example of a symmetric matrix is the identity matrix, which has 1s along the main diagonal and 0s elsewhere. Another example is the matrix [1 2; 2 3], where the elements on either side of the main diagonal are equal.

## 4. Is every square matrix symmetric?

No, not every square matrix is symmetric. A matrix is considered symmetric if it is equal to its transpose. Therefore, a matrix that is not equal to its transpose is not symmetric.

## 5. Are there any other methods for proving symmetry of a matrix?

Yes, there are other methods for proving symmetry of a matrix, such as using the properties of transposition and proving that the matrix is equal to its negative. However, the method of showing that (A + A(Transpose))(Transpose) = A + A(Transpose) is the most commonly used and straightforward method for proving symmetry.

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