Cat causing rash??

My DS has had exma since I stopped BF at 2 1/2 months,he is now 9 months. I tried eveything I can think of, swapping washing powders to sensitive ones, changing all bath products, being careful with foods, trying different formulas, even a special naturapath diet. I was wondering if anyone might know if its possible that he could be allergic to our cat? I am allergic to cats and I only just realised this is what might be causing it. We have alot of carpet too which I try and keep the cat off but DS crawls after the cat and pulls out bunches of fur so he is having direct contact with him now. He also alot of the time has a dry cough that doesnt seem to bother him and I always wondered if it was fake or not.
Has anyone heard of cats causing this kind of eczema reaction in infants?