1) If A is an n by n matrix then det(adj A) = (det A)^(n-1). Also, if det A is not zero then A = (det A)(adj A)^(-1). For this example,
det(adj A) , and so det A = 2. You then have to find the inverse of adj A and multiply it by 2, in order to get A.
2) A vector space has to have a zero element. If R with the given operation is a vector space then there must exist an element O of R such that x Å O = x for every x in R. Think about what that says about O. (Of course, O need not be the usual 0. In fact, it would have to have very unusual properties.)