Question

Why is matrix addition associative?

    Why is matrix addition associative?

Finite Mathematics and Applied Calculus

Finite Mathematics and Applied Calculus

Stefan Waner, Steven… 5th Edition

Chapter 3, Problem 63

Instant Answer

Solved by Expert Mahendra Kumar

09/02/2021

Step 1

In other words, for matrices A, B, and C, the following equation holds true: (A + B) + C = A + (B + C) To understand why this is the case, let's consider the definition of matrix addition. When you add two matrices, you add their corresponding elements. So, for Show more…

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Why is matrix addition associative?

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Key Concepts

Component-wise Operation

Since matrix addition is carried out entry by entry, its associativity is inherited from the associativity of addition in the underlying number system. Because the sum of the corresponding entries is independent of how the numbers are grouped, the overall operation of matrix addition is also associative.

Matrix Addition

Matrix addition is defined as the element-wise addition of two matrices of the same dimensions. Each entry in the resulting matrix is the sum of the corresponding entries from the two matrices, making it an operation that relies on the underlying arithmetic of the field from which the entries are taken.

Associative Property

The associative property is a fundamental algebraic rule which states that for any three elements, the grouping of operations does not affect the result. In the context of matrix addition, this means that when adding three matrices, the sum obtained by adding the first two and then adding the third is the same as adding the second two and then the first, i.e., (A+B)+C = A+(B+C).

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