Symmetric assumption in linear regression

Question

I have referred to https://stats.stackexchange.com/questions/347482/differentiation-of-rss, however there is an implicit assumption about : $(Y^T)$(XB) = $(XB)^T$Y. Why do we assume the product of the two matrices to be symmetric. I am not sure about the intuition behind that.

Answer 1

Observe that both $y^TX\beta$ and $(X\beta)^Ty$ are just numbers ($1\times 1$ matrices). Now you have $$ (X\beta)^Ty=\beta^TX^Ty=\beta^T(y^TX)^T=(y^TX\beta)^T $$ So your two real numbers are the transpose of each other, so they are equal.