Set S of Square Matrices with Respect to Matrix Addition
Matrix Addition in Set S:
- The set S consists of square matrices of the same order.
- Matrix addition in set S is defined as adding corresponding elements of two matrices of the same order.
Closure Property:
- The set S is closed under matrix addition.
- When two matrices from set S are added, the result is also a square matrix of the same order.
- This property ensures that the operation of matrix addition is well-defined within set S.
Identity Element:
- In set S, the identity element for matrix addition is the zero matrix.
- The zero matrix is a square matrix with all elements being zero.
- When the zero matrix is added to any matrix in set S, the result is the original matrix.
Associative Property:
- Matrix addition in set S follows the associative property.
- For three matrices A, B, and C in set S, (A + B) + C = A + (B + C).
- This property allows for the grouping of matrices in any order during addition without changing the result.
Commutative Property:
- Matrix addition in set S also satisfies the commutative property.
- For two matrices A and B in set S, A + B = B + A.
- This property ensures that the order of matrices being added does not affect the result.
Overall, the set S of square matrices with respect to matrix addition forms a closed algebraic structure with well-defined properties, making it a fundamental concept in linear algebra.