Chapter 7.2, Problem 31E

Exercises 23 and 24 show how to classify a quadratic form Q(x) = x^TAx, when $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ and det A ≠ 0, without finding the eigenvalues of A.

23. If λ₁ and λ₂ are the eigenvalues of A, then the characteristic polynomial of A can be written in two ways: det(A − λI) and (λ − λ₁)(λ − λ₂). Use this fact to show that λ₁ + λ₂ = a + d (the diagonal entries of A) and λ₁λ₂ = det A.