Chapter 5, Problem 17SE

Let A = $\left[\begin{array}{cc}{\text{a}}_{11}& {\text{a}}_{12}\\ {\text{a}}_{21}& {\text{a}}_{22}\end{array}\right]$. Recall from Exercise 25 in Section 5.4 that tr A (the trace of A) is the sum of the diagonal entries in A. Show that the characteristic polynomial of A is

λ² — (tr A)λ + det A

Then show that the eigenvalues of a 2 × 2 matrix A are both real if and only if $det\text{A}\le {\left(\frac{tr\text{A}}{2}\right)}^2$.