# Matrix Entrywise Addition over Ring is Associative

## Theorem

Let $\struct {R, +, \circ}$ be a ring.

Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $R$.

For $\mathbf A, \mathbf B \in \map {\MM_R} {m, n}$, let $\mathbf A + \mathbf B$ be defined as the matrix entrywise sum of $\mathbf A$ and $\mathbf B$.

The operation $+$ is associative on $\map {\MM_R} {m, n}$.

That is:

$\paren {\mathbf A + \mathbf B} + \mathbf C = \mathbf A + \paren {\mathbf B + \mathbf C}$

for all $\mathbf A$, $\mathbf B$ and $\mathbf C$ in $\map {\MM_R} {m, n}$.

## Proof 1

Let $\mathbf A = \sqbrk a_{m n}$, $\mathbf B = \sqbrk b_{m n}$ and $\mathbf C = \sqbrk c_{m n}$ be elements of the $m \times n$ matrix space over $R$.

Then:

 $\ds \paren {\mathbf A + \mathbf B} + \mathbf C$ $=$ $\ds \paren {\sqbrk a_{m n} + \sqbrk b_{m n} } + \sqbrk c_{m n}$ Definition of $\mathbf A$, $\mathbf B$ and $\mathbf C$ $\ds$ $=$ $\ds \sqbrk {a + b}_{m n} + \sqbrk c_{m n}$ Definition of Matrix Entrywise Addition over Ring $\ds$ $=$ $\ds \sqbrk {\paren {a + b} + c}_{m n}$ Definition of Matrix Entrywise Addition over Ring $\ds$ $=$ $\ds \sqbrk {a + \paren {b + c} }_{m n}$ Ring Axiom $\text A1$: Associativity of Addition $\ds$ $=$ $\ds \sqbrk a_{m n} + \sqbrk {b + c}_{m n}$ Definition of Matrix Entrywise Addition over Ring $\ds$ $=$ $\ds \sqbrk a_{m n} + \paren {\sqbrk b_{m n} + \sqbrk c_{m n} }$ Definition of Matrix Entrywise Addition over Ring $\ds$ $=$ $\ds \mathbf A + \paren {\mathbf B + \mathbf C}$ Definition of $\mathbf A$, $\mathbf B$ and $\mathbf C$

$\blacksquare$

## Proof 2

By definition, matrix entrywise addition is the Hadamard product of $\mathbf A$ and $\mathbf B$ with respect to ring addition.

We have from Ring Axiom $\text A1$: Associativity of Addition that ring addition is associative.

The result then follows directly from Associativity of Hadamard Product.

$\blacksquare$