Answer to Question #332135 in Linear Algebra for peac_eboy

Question

1.1 Prove that the characteristic polynomial of a 2 × 2 matrix A can be expressed as

λ^2 − tr(A)λ + det(A), where tr(A) is the trace of A.

Answer 1

Let $A=\begin{pmatrix} a&b\\c&d \end{pmatrix}$ . Then $\det (A)=ad-bc$ and $\text{tr}(A)=a+d$ .

The characteristic polynomial: $\det(A-\lambda I)=\begin{vmatrix}a-\lambda &b\\c&d-\lambda \end{vmatrix}=(a-\lambda)(d-\lambda )-bc=\lambda^2-(a+d)\lambda +ad-bc=\lambda^2-\text{tr}(A)\lambda +\det(A)$