Which one of the following statements is true for all real symmetric matrices? A. All the eigen values are real B. All the eigen values are positive C. All the eigen values are distinct D. Sum of all the eigen values is zero

The correct answer is A. All the eigenvalues are real.

A real symmetric matrix is a square matrix that is equal to its transpose. The eigenvalues of a real symmetric matrix are always real numbers. This is because the eigenvalues of a matrix are the roots of its characteristic polynomial, which is a polynomial with real coefficients.

The other options are not always true for real symmetric matrices. For example, the eigenvalues of a real symmetric matrix can be negative. This is because the characteristic polynomial of a real symmetric matrix can have real negative roots.

The sum of all the eigenvalues of a real symmetric matrix is not always zero. This is because the characteristic polynomial of a real symmetric matrix can have real nonzero roots.