Someone (Ph.D in Maths) told me that matrix addition(yes addition) is not comutative.
but how it is possible. He is wrong or right. If he is right then how?
I'm 12 std. Student
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Prashant
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6Matrix addition is always commutative. It is the multiplication that is not. – marwalix Aug 28 '16 at 07:44
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5I’m voting to close this question because the only interesting aspect of it is that a PhD in math made such a false claim. – Kurt G. Nov 19 '23 at 08:35
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2? The OP wrote matrix addition. – Anne Bauval Nov 22 '23 at 09:20
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@DanieleTampieri . As Anne Bauval pointed out and to be precise, OP wrote "matrix addition *(yes addition)*". – Kurt G. Nov 22 '23 at 09:30
2 Answers
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Interpreting $n\times n$-matrices by vectors of length $n^2$, matrix addition is just addition of vectors in a vector space $V$. Since $(V,+)$ is a commutative group (by definition of a vectors space), the addition is commutative.
Dietrich Burde
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Addition of matrices is only defined for matrices with the same rows and colums. For two matrices $\mathbf{A} = \{ a_{ij} \}$ and $\mathbf{B} = \{ b_{ij} \}$ with the same rows and colums we have that $\mathbf{A} + \mathbf{B} = \{ a_{ij} + b_{ij} \}$. Therefore:
$$\mathbf{A} + \mathbf{B} = \{ a_{ij} + b_{ij} \} = \{ b_{ij} + a_{ij} \} = \mathbf{B} + \mathbf{A}$$
So matrix addition is commutative. For a reference you can check Chapter 1, equation 2.2 from:
- Harville, David A. Matrix Algebra from a Statistician’s Perspective. Springer, 2008.
luifrancgom
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