How to find characteric polynomial of a 2x2 matrix?

The roots of the characteristic equation are the eigenvalues. Suppose they are a and b, then the characteristic equation is (x-a) (x-b) = 0⟹x2 - (a+b) x+ab=0. The product of eigenvalues is the determinant, and the sum is the trace, hence we get the equation x2-tr (A) x+det (A) = 0.