4.4.2

a) Let A be a real symmetric 2x2 matrix, $$ A = \begin{bmatrix} x & y \\ y & z \\ \end{bmatrix}$$. The eigenvalues $$\lambda $$ of A are the roots of the characteristic polynomial, so they solve $$\lambda^2 - (tr \ A)\lambda + det \ A = 0$$, or $$ \lambda^2 - (x+z)\lambda + (xz - y^2) = 0$$.

The discriminant of this quadratic equation is $$ D = (x+z)^2 - 4(xz-y^2) = (x-z)^2 + 4y^2 \geq 0 $$, since x,y and z are all real, thus the roots of the quadratic are real numbers, and so are the eigenvalues of A.

b) Let A be a real 2x2 matrix with positive off diagonal entries, $$A = \begin{bmatrix} w & x \\ y & z \\ \end{bmatrix}, \ x,y > 0 $$. As before, the eigenvalues $$\lambda $$ satisfy the quadratic equation $$\lambda^2 - (w+z) + (wz - xy) = 0$$, which has discriminant $$ D = (w+z)^2 - 4(wz-xy) = (w-z)^2 + 4xy > 0 $$, so the roots of this quadratic are real, and so are the eigenvalues of A.