MHB Commutativity in the linear transformation space of a 2 dimensional Vector Space

[quarkine]

A variant of a problem from Halmos :
If AB=C and BA=D then explain why (C-D)^2 is commutative with all 2x2 matrices if A and B are 2x2 matrices.
This result does not hold for any other nxn matrices where n > 2. Explain why.

Edit: I tried to show that ((C-D)^2) E - E((C-D)^2) is identically zero. But that didn't work.
A guess is that since the above matrix commute with any 2x2, it has to be of the form bI (where b is a scalar anad I the indentity), which can be confirmed by brute calculation but I am searching for a better way.

 

[MarkFL]

Can you give your thoughts and/or show what you have tried so our helpers know exactly where you are stuck and how best to help?

 

[Opalg]

If AB=C and BA=D then explain why (C-D)^2 is commutative with all 2x2 matrices if A and B are 2x2 matrices.

This is known as Hall's identity. The proof, in very brief outline, goes like this. The commutator $[A,B] = AB-BA$ has trace zero. Its characteristic equation is therefore of the form $\lambda^2 = \mathrm{const.}$ It then follows from the Cayley–Hamilton theorem that $[A,B]^2$ is a multiple of the identity.