Explain why a 2x2 matrix can have at most two distinct eigenvalues

Hey there. Hope I can help.

So lets say that a 2 * 2 matrix (A) has three distinct eigenvalues.

You need to remember that the eigen vectors state  which correspond to the distinct Eigen values of an n * n matrix, then our set is linearly independent. 

Which we can now tell by this theorem that three linearly independent eigenvectors correspond which is literally absurd.

The reason for this is because the matrix A is only two dimensional which means the three vectors belong to R^2. So any set in R^n would be linearly independent if r > n since (r = 3) > (n = 2). These three eigenvectors then become linearly independent. Therefore the 2 * 2 matrix can only have 2 atmost.