Group Theory in Mathematics:

Group theory is a branch of abstract algebra that deals with the study of groups. A group is a mathematical structure consisting of a set of elements and a binary operation that combines two elements to produce a third element. The binary operation must satisfy certain conditions, such as associativity, identity, and inverse.

Examples of Groups:

There are many examples of groups in mathematics, such as:

- The set of integers under addition
- The set of real numbers under addition
- The set of non-zero complex numbers under multiplication

Non-examples of Groups:

However, not all sets with a binary operation form a group. For example:

- The set of all rational numbers under addition does not form a group because it does not contain an identity element.
- The set of all integers under multiplication does not form a group because not all elements have an inverse.

Answer Explanation:

Option B is true, which states that the set of all non-singular matrices forms a group under multiplication. Let's see why this is true.

Definition of Non-singular Matrices:

A non-singular matrix is a square matrix that has a determinant, which is a scalar value that can be computed from the entries of the matrix. If the determinant is non-zero, the matrix is said to be non-singular. Otherwise, it is singular.

Properties of Non-singular Matrices:

- Non-singular matrices are invertible, which means that there exists another matrix that, when multiplied by the original matrix, results in the identity matrix.
- The product of two non-singular matrices is also non-singular.
- The inverse of a non-singular matrix is unique.

Proof that Non-singular Matrices form a Group:

To show that the set of all non-singular matrices forms a group under multiplication, we need to verify the following properties:

1. Closure: The product of two non-singular matrices is also non-singular.

2. Associativity: The product of three or more non-singular matrices is independent of the order of multiplication.

3. Identity element: There exists a non-singular matrix I such that AI = IA = A for all non-singular matrices A.

4. Inverse element: For every non-singular matrix A, there exists a unique non-singular matrix A^-1 such that AA^-1 = A^-1A = I.

Closure:

Let A and B be two non-singular matrices. We need to show that AB is also non-singular.

Suppose AB is singular, which means that det(AB) = 0. Then det(A)det(B) = det(AB) = 0. Since A and B are non-singular, det(A) ≠ 0 and det(B) ≠ 0. Therefore, it must be the case that det(AB) ≠ 0, which contradicts our assumption. Hence, AB is non-singular.

Associativity:

Let A, B, and C be non-singular matrices. We need to show that (AB)C = A(BC).

(AB)C = A(BC) = ABC

Identity element:

Let I be the identity matrix. We need to show that AI = IA = A for all non-singular matrices A.

AI = A and IA = A because the product of any matrix and the identity matrix is the original matrix.

Inverse element:

Let A be a non-singular matrix. We need to show that there exists