The identity element in multiplication Let's get on with the question. Is it true that the multiplication is always commutative when we multiply an entity (number, matrix etc.) with the identity entity? I mean, the statement is true in the scope of numbers ($a×1=1×a=a$) and even in the scope of matrices , where multiplication is not commutative from general perspective.
If it is true, then what would be the proof of it? (an as general as possible proof). The smallest but strong starting point of it would be of great use. Thanks in advance!

An identity need not commute with every entity. There are left identities and right identities. For example, with 2 elements $a$ and $b$, define $a \times b = b$ and $b \times a = a$. In this case both $a$ and $b$ are left identities, and there is no defined right identity. If $a$ and $b$ happen to equal each other, then we can call it a two-sided identity, which is what we usually deal with when we have identities--just not always.
