curious on matrices

Why is it that if a 2 by 2 matrix can commute with both
(1 0
0 0) and
(1 1
0 0)
then it can commute with every 2 by 2 matrix?

What have you tried?

oh i discovered that in order to commute with
(1 0
0 0) then it has to be diagonal matrix
in order to commute with
(1 1
0 0) you have to have a matrix where
(a b
0 a-b)
i set them equal to each other and I got general 2 by 2 matrix
(a 0
0 a) and i found out this can commute with every 2 by 2.
This isn't a coincidence right?

Well
(a 0
0 a)

is simply "a" times the identity matrix, hence the commutability (if that's a real word).

Well
(a 0
0 a)

is simply "a" times the identity matrix, hence the commutability (if that's a real word).

ahh so theres nothing to do with the fact that it commutes with the other two matrices. thank!

Well there is. The fact that a matrix commutes with the other two implies that it is of the form $aI$ for some real a.