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.
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.
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.
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.
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.