Matrix addition is comutative or not

Question

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

Answer 1

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.

Answer 2

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.