Is the set of 4 times 4 invertible matrices a subspace of R4

Is the set of 4 times 4 invertible matrices a subspace of R^4 x d?

Solution

Let A and B be 2 arbitrary elements of the set S (say) of 4 x 4 invertible matrices. Since A+B need not be invertible, when A and B are invertible matrices, and since, in general (A+B)^-1 A^-1 + B^-1. Hence S is not closed under vector addition. Hence S is not a vector space.Hence S is not a subspace of R^4x4.