Question

8. Which sentence is true?
A. Set of all matrices forms a group under multiplication
B. Set of all rational negative numbers forms a group under multiplication
C. Set of all non-singular matrices forms a group under multiplication
D. Both (b) and (c)​

Answer 1

C. Set of all non-singular matrices forms a group under multiplication

• A group is a collection of items that can be combined into a third element by any two other elements in the group. A group must also meet the following requirements:
• Any two components in the set can be manipulated openly by an operator to create a third element, which must also be in the set. This is known as the closure property.
• The order of operation is irrelevant for an expression with three or more operands that share the same operator as long as the order of the operands is left unchanged. As an illustration, a + (b + c) equals (a + b + c).
• Each set is required to contain an identity element, which is an element that, when combined with another element in the set, produces the element itself. For illustration, a + 0 = a. The identity element in this case is 0.
• The set should have an inverse for each element, according to the invertibility property.
• For the aforementioned claims, we now have
• B is untrue because it violates the closure property. Two negative numbers multiplied together yield a positive number that is not part of the set.
• C is accurate. For the inverse to exist, the matrices in the set must not be singular, that is, their determinant must not be zero (Invertibility Property).
• A is erroneous since there is no such thing as a singular (determinant = 0) matrix's inverse (Invertibility Property violated).

Hence, option C is correct.

Learn more here

https://brainly.in/question/7230027

#SPJ1