commutativity of non-scalar matrices Problem:
Prove that the commutativity on the set of non-scalar $2\times2$ matrices is an equivalence relation. (That is, for all A; B; and
C; if AB = BA and BC = CB then AC = CA:)
For commutativity to be equivalence relation, we have to it is reflexive, symmetric and transitive. The first two properties are obvious. Any help on how to prove the third property?

This is a write-up of the comments adding some thoughts that came to my mind:


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*The first step is to show that $A$ and $B$ commute iff $S^{-1}AS$ and $S^{-1}BS$ commute. Therefore it follows that for the proof of transitivity, i.e. that we have $A$, $B$ and $C$ with $A$ and $B$ commuting and $B$ and $C$ commuting, we can assume that $B$ is in Jordan normal form. Now because we look at two cases: 2./3. (since $B$ is not a scalar multiple of the identity. 

*The matrix $B$ is a diagonal matrix with two distinct eigenvalues. In this case, as the OP already remarked it follows that also $A$ and $C$ are diagonal matrices and hence they commute.

*The matrix $B$ consists of one Jordan block to a single eigenvalue. In this case, both $A$ and $C$ also have to be Jordan blocks with a single eigenvalue and one checks by direct computation that these commute.

