I am trying to show that converse of the statement: Two 2 by 2 matrices over F which are not scalar matrices are similar if and only if they have the same characteristic polynomial.
Here is my attempt:
If the polynomial has two distinct roots in F, clearly two are similar to the diagonal matrix.
If the polynomial only has a repeated root, then two are similar to the elementary Jordan matrix since these two are not scalar matrices.
If the polynomial doesn't have roots in F, then ..........
Thanks