Is any matrix a tensor in special relativity? My question is inspired by the definition of the electromagnetic field tensor in Carroll's Spacetime and Geometry book. In equation (1.69) he defines a matrix which he says is the (0,2) electromagnetic field tensor. In section 1.8 he states the Maxwell equations in tensor notation and states that in this form they manifestly transform as tensors. I assume that this is due to the tensorial nature of the electromagnetic field tensor, however I don't understand why it is a tensor. Would we have to prove that the electromagnetic field tensor transforms as a tensor by brute calculation or is there some cleverer, more obvious shorthand to come to the same realization?

Thanks