I am confused about the definition of subgroups.

I am looking at matrices of the form [a 0 ; 0 0] with a not equal to 0. This is a subset of GLn2

I thought this was a subgroup but my book says it's not. I figured that any matrix of that form multiplied by [1 0; 0 0] gives back the matrix, so [1 0; 0 0] is the identity and is in the group. Also, any matrix of that form multiplied [1/a 0; 0 0] gives the previously found identity.

Does the identity of the subgroup need to be the identity of the larger group (in this case [1 0; 0 1]) or the identity that works for the subgroup?