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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

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If the polynomial doesn't have roots in $F$, then their rational canonical form is the same: the companion matrix of the characteristic polynomial, hence they are similar.