GATE | GATE-CS-2002 | Question 6
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 matrices forms a group under multiplication.
(D) Both (2) and (3) are true.
Explanation: A group is a set of elements such that any two elements of the group combine to form a third element of the same group. Also, a group must satisfy certain properties:
Closure Property – Any two elements of the set when operated open by an operator form a third element that must also be in the set.
Associative Property – For an expression with three or more operands having the same operator between them, the order of operation does not matter as long as the sequence of operands are not changed. For example, (a + b) + c = a + (b + c).
Identity element Property – Each set must have an identity element, which is an element of the set such that when operated upon with another element of the set, it gives the element itself. For example, a + 0 = a. Here, 0 is the identity element.
Invertibility Property – For each element of the set, inverse should exist.
Now, for the given statements, we have
A is incorrect as it does not satisfies closure property. If we take two negative numbers and multiply them, we get a positive number which is not in the set.
B is correct. The matrices in the set must be non – singular, i.e., their determinant should not be zero, for the inverse to exist (Invertibility Property).
C is incorrect as the inverse of a singular (determinant = 0) matrix does not exist (Invertibility Property violated).
Thus, B is the correct option.
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