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The addition of matrices is an operation allowing us to add matrices as we can do with numbers, i.e. instead of calculating the result of matrix addition component by component, it is possible to significantly abbreviate the notation of such an addition using just one symbol for the matrices we want to add together. The following proposition justifies this abbreviation:

Proposition: Abelian Group of Matrices Under Addition

Let $F$ be a field. The set of all $m\times n$ matrices $M_{m\times n}(F)$ together with the matrix addition $”+”$ constitutes an Abelian group, i.e. the set $(M_{m\times n}(F), +)$ fulfills the following laws:

Associativity $A+(B+C)=(A+B)+C$ for all matrices $A,B,C\in M_{m\times n}(F)$.

Commutativity $A+B=B+A$ for all matrices $A,B\in M_{m\times n}(F)$.

The zero matrix $O\in M_{m\times n}(F)$ is the neutral element: $O+A=A+O=A$ for all $A\in M_{m\times n}(F)$.

For all matrices $A\in M_{m\times n}(F)$ there is a matrix $-A$ such that $A+(-A)=(-A)+A=O.$