3Blue1Brown has a nice new video on how to calculate the eigenvalues of 2×2 matrices.
The most common way to find the eigenvalues of a 2×2 matrix A is working straight from the definition, solving det(A – λI) = 0.
This is fine when you’re learning what eigenvalues are. But if you’ve already learned all the theory and just want to calculate the eigenvalues, there’s an easier way.
As shown in the video, the eigenvalues are
m ± √(m² – p)
where m is the mean of the elements on the main diagonal and p is the determinant.