Which of the following is true? A. The set of all rational negative numbers forms a group under multiplication. B. The set of all non-singular matrices forms a group under multiplication. C. The set of all 2 x 2 matrices forms a group under multiplication. D. Both B and C are true.
GATE 2002 · Discrete Mathematics · Group Theory · medium
Answer: Only the set of non-singular matrices satisfies all group axioms, so the answer is B.
- Reject A and C by axiom failures: A: the product of two negative rationals is positive, so the set is not closed - not a group. C: the set of all 2x2 matrices contains singular matrices (e.g. the zero matrix) that have no multiplicative inverse, so the inverse axiom fails - not a group. Hence D ('both B and C') is also false because C is false
- Confirm B: non-singular matrices form a group: the product of two non-singular matrices is non-singular (det(AB) = det(A)det(B) != 0), matrix multiplication is associative, the identity matrix I is non-singular and acts as identity, and every non-singular matrix has an inverse by definition - all four axioms hold, so the non-singular matrices form the general linear group