Proof of Square Matrices Form a Group Under Addition: Explained

Matrix addition is a fundamental operation in linear algebra with wide-ranging applications in mathematics, physics, and computer science. Specifically, the concept of square matrices forming a group under the addition operation is a crucial aspect of group theory. In this article, we will delve into the proof and provide a comprehensive understanding of this concept.

Introduction to Matrix Addition

Matrix addition involves combining the corresponding entries of two matrices to produce a new matrix. For the matrices to be added, they must be of the same dimensions. However, the concept extends beyond square matrices to include non-square matrices of the same size. In this discussion, we will focus on square matrices, as they are more common and often used in theoretical contexts.

Matrices and Fields

The entries of matrices are typically taken from a field such as the rational numbers (mathbb{Q}), real numbers (mathbb{R}), or complex numbers (mathbb{C}). A field is an algebraic structure connected with the operations of addition, subtraction, multiplication, and division, which are well-defined and satisfy certain properties such as associativity, commutativity, and distributivity.

Associativity of Matrix Addition

One of the essential properties of a group is associativity. In the context of matrices, associativity means that for any three matrices A, B, and C, the following equation holds:

$$(A B) C A (B C)$$

This property can be verified for each corresponding entry of the matrices. Consider the element at the i-th row and j-th column of the matrices A, B, and C. Let these entries be denoted as (a_{ij}), (b_{ij}), and (c_{ij}), respectively. Then, by the definition of matrix addition, we have:

$$(a_{ij} b_{ij}) c_{ij} a_{ij} (b_{ij} c_{ij})$$

Given that addition in the underlying field (mathbb{Q}, mathbb{R}, mathbb{C}) is associative, this equation holds true for all i and j. Therefore, matrix addition is associative.

Identity Element of Matrix Addition

The identity element in the context of matrix addition is the zero matrix, denoted as O. The zero matrix has all entries equal to zero. For any matrix A, the sum of A and the zero matrix is equal to A:

$$A O O A A$$

This means that adding the zero matrix to any matrix does not change the original matrix. Hence, the zero matrix serves as the identity element in the group of matrices under addition.

Inverse Elements in Matrix Addition

For a matrix A, its inverse under matrix addition is -A, which is the matrix with each entry negated. For any matrix A, the sum of A and -A is the zero matrix:

$$A (-A) (-A) A O$$

This means that for every matrix A, there exists an entry-wise negation of the matrix, which acts as the additive inverse. This property is a fundamental requirement for the elements of a group.

Commutativity of Matrix Addition

Another important property of the group of matrices under addition is commutativity. For any two matrices A and B, the sum A B is equal to B A:

$$A B B A$$

Similar to the associativity proof, the commutativity can be verified using the corresponding entries of the matrices. For any i and j:

$$a_{ij} b_{ij} b_{ij} a_{ij}$$

Given that addition in the underlying field is commutative, this equation holds for all i and j. Therefore, matrix addition is commutative.

Conclusion

In summary, the set of square matrices, along with the addition operation, form a group. This group is closed under addition, has an identity element (the zero matrix), for each matrix A there exists an inverse (-A), and the addition is both associative and commutative. Understanding these properties is fundamental for grasping the broader concepts in linear algebra and abstract algebra.

Further Exploration

To deepen your understanding, you can start with simpler cases like 1x1 matrices (which are simply scalars) or 1x2 matrices. These cases can provide insights into the properties of matrix addition before moving on to more complex square matrices.