Eigenvalues of a Symmetric 2x2 Matrix

Solve $\det (A-\lambda I)=0$.

The characteristic polynomial is $(2-\lambda {)}^2-1=0$.

1. 1. Set up the characteristic equation

   The determinant of $A-\lambda I$ must vanish.

   $$(2-\lambda {)}^2-1=0$$

2. 2. Solve for lambda

   Expanding gives ${\lambda }^2-4\lambda +3=0$, which factors as $(\lambda -3)(\lambda -1)=0$, so the eigenvalues are $3$ and $1$.

Answer: 3 and 1