How do I calculate the eigenvalues and eigenvectors of a 2x2 matrix, and what is the point of doing this calculation?

We find the eigenvalues (here called "k") by solving the characteristic equation det(M - kI) = 0. For a 2x2 matrix ((a, b), (c,d)) the determinant is ad - bc, we set this equal to zero and solve the resulting quadratic (using the quadratic formula or otherwise). We can then substitute the found values of k into the eigenvalue equation Mv = kv to find the eigenvectors v by observing that (M - kI)v = 0 and solving the resulting system of equations.The use of these calculations is that they completely characterise the action of the matrix in question. From the eigenvalues and eigenvectors we can see exactly how a matrix affects other objects. This is especially useful in fields such as physics where we want to use mathematics to model the world around us.