Let the set of all 2 x 2 matrices. Let the operation of vector addition in V be the usual matrix addition but let scalar multiplication in V be defined by:

Show that V is not a vector space by demonstrating with an example that one of the properties in the definition of a vector space fails to hold.

My attempt:

property I believe fails to hold (let r be any real number):

I'll define both A and B as the 2 x 2 matrices:

and

=> => =>

Hence, V is not a vector space since it does not hold for the distributivity of scalar multiplication with respect to vector addition. Did I go about this proof correctly?