Find out if each of the following is vector space. If it is the vector space, then prove that it is a vector space. If it isn't a vector space, then give an example of where it fails to be a vector space:
Let GL(n) be the set of nonsingular n × n matrices. For any two matrices A and B in GL(n) define A [+] B = AB where AB is matrix multiplication. For any real number r and nonsingular matrix A, define r [*] A = Ar.