Remember the four defining properties of a group: closure, associativity, identity element, and inverse:
1. Closure - For every pair of elements a and b, a*b must be an element of the group.
2. Associativity - For any three elements a, b, and c, the equality a*(b*c)=(a*b)*c must hold.
3. Identity element - There is a group element e, the identity element, such that a*e=e*a=a for any a in the group.
4. Inverse - For every group element a, there is an element b, the inverse, such that a*b=b*a=e, where e is the identity.
You just need to show that the set of all invertible 2x2 matrices with real
entries under matrix multiplication fits these. For example, the existence of the 2x2 identity matrix establishes #3, the matrices being invertible satisfies #4.