Let A be the set of all non-singular matrices over real numbers and let * be the matrix multiplication operation. Then A. A is closed under * but <A, *> is not a semigroup. B. <A, *> is a semigroup but not a monoid. C. <A, *> is a monoid but not a group. D. <A, *> is a group but not an abelian group.

GATE 1995 · Discrete Mathematics · Group Theory · medium

Answer: D. <A, *> is a group but not an abelian group

1. Closure and associativity: Product of two non-singular matrices has nonzero determinant, so it stays in A; matrix multiplication is always associative.
2. Identity and inverses: I is non-singular (det = 1), and every non-singular A has an inverse A^(-1) which is also non-singular.
3. Test commutativity: A concrete pair of invertible matrices fails to commute, so the group is non-abelian.