If a square matrix A is real and symmetric, then the eigen values A. are always real B. are always real and positive C. are always real and non-negative D. occur in complex conjugate pairs

The correct answer is: A. are always real.

A square matrix A is real and symmetric if $A^T = A$. The eigenvalues of a real symmetric matrix are always real. This is because the characteristic polynomial of a real symmetric matrix is a real polynomial, and the roots of a real polynomial are always real.

The characteristic polynomial of a square matrix $A$ is defined as $p(x) = |xI – A|$. The eigenvalues of $A$ are the roots of $p(x)$.

If $A$ is real and symmetric, then $p(x)$ is a real polynomial. This is because $xI – A$ is a real matrix, and the determinant of a real matrix is always real.

The roots of a real polynomial are always real. This is because the Fundamental Theorem of Algebra states that every non-constant single-variable real polynomial has at least one real root.

Therefore, the eigenvalues of a real symmetric matrix are always real.