Let $a, b, c ∈ R$, and let $M$ be the following matrix. Prove that $(a + c)^2 − 4ac ≥ 0$.

Question

Let $a, b, c ∈ R$, and let $M$ = \begin{pmatrix} a & b \\ b & c \end{pmatrix}

Prove that M has a real eigenvalue

Answer 1

Hint: The characteristic equation is $$\lambda^2 -(a+c)\lambda + \left(ac-b^2\right) = 0.$$

What is the discriminant of the quadratic here (in particular its sign), and what does that tell you about the roots?