Answer: so the addition is not a problem here, i understand that you write a*A = a.A so is a scalar product. if your matrix is A1, A2, A3, A4, and you multiply this with an scalar a defined in the real set.
then a*A = a.A1, a.A2, a.A3, a.A4
so this multiplication still gives you a 2x2 matrix, that is in your space.
so the defined here is a Vector space, because the addition of two of his objects results in other object in the set.
And also has a scalar multiplication that if you feed it with an object on the set, it returns another object on the set.
