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 [tex]V_1,...,v_r[/tex] which correspond to the distinct Eigen values [tex]λ_1,...,λ_n[/tex] of an n * n matrix, then our set [tex]{V_1,...,v_r}[/tex] 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 [tex]{V_1,...,v_r}[/tex] 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.