There is a question here on Stackexchange- Does the order of explanatory variables matter when calculating their regression coefficients?
Although I understand the details here, how can we formally prove that the order of variables doesn't change the coefficient estimates? I was trying to use partitioned matrices, and just showing that by changing the order of partitions and the respective coefficient estimates matrix, I get the same result. However, it doesn't look like a formal proof. Can anyone help?