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Matrix Addition Is Comutative Or Not [closed]

4 min read Sep 25, 2024
Matrix Addition Is Comutative Or Not [closed]

In linear algebra, matrix addition is a fundamental operation that involves combining two matrices of the same dimensions. One of the key properties often investigated is whether matrix addition is commutative, meaning whether the order in which matrices are added affects the resulting sum. This article explores the nature of matrix addition and investigates whether it adheres to the commutative property.

Understanding Matrix Addition

Matrix addition is a straightforward process involving adding corresponding elements of two matrices. For instance, if we have two matrices, A and B, both with dimensions m x n, their sum, denoted as A + B, is another m x n matrix where each element is the sum of the corresponding elements in A and B.

Example:

Let's consider two matrices:

A =  [ 1 2 ]
     [ 3 4 ]

B =  [ 5 6 ]
     [ 7 8 ]

Their sum, A + B, would be:

A + B = [ 1 + 5  2 + 6 ]
          [ 3 + 7  4 + 8 ]

         = [ 6 8 ]
           [ 10 12 ]

Commutativity in Matrix Addition

To determine if matrix addition is commutative, we need to examine whether A + B is equal to B + A for any arbitrary matrices A and B.

Let's assume we have two matrices A and B, both with dimensions m x n. We can represent their elements as follows:

  • A = [a<sub>ij</sub>]
  • B = [b<sub>ij</sub>]

Where i represents the row number and j represents the column number.

Calculating A + B:

A + B = [aij + bij]

Calculating B + A:

B + A = [bij + aij]

Since addition is commutative for real numbers, we have:

aij + bij = bij + aij

Therefore, the elements of both A + B and B + A are identical, proving that:

A + B = B + A

Conclusion

The analysis above demonstrates that matrix addition is indeed commutative. The order in which matrices are added does not alter the final result. This property holds true for any two matrices with compatible dimensions. The commutativity of matrix addition is an important concept in linear algebra, simplifying calculations and enabling various manipulations within matrix operations.

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