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.