# Matrix Entrywise Addition is Associative

## Theorem

Let $\map \MM {m, n}$ be a $m \times n$ matrix space over one of the standard number systems.

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

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

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 $\map \MM {m, n}$.

## Proof 1

From:

Integers form Ring
Rational Numbers form Ring
Real Numbers form Ring
Complex Numbers form Ring

the standard number systems $\Z$, $\Q$, $\R$ and $\C$ are rings.

Hence we can apply Matrix Entrywise Addition over Ring is Associative.

$\Box$

The above cannot be applied to the natural numbers $\N$, as they do not form a ring.

However, from Natural Numbers under Addition form Commutative Monoid, the algebraic structure $\struct {\N, +}$ is a commutative monoid.

The result follows from Associativity of Hadamard Product.

$\blacksquare$

## Proof 2

Let $\mathbf A = \sqbrk a_{m n}$, $\mathbf B = \sqbrk b_{m n}$ and $\mathbf C = \sqbrk c_{m n}$ be matrices whose order is $m \times n$.

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 $\ds$ $=$ $\ds \sqbrk {\paren {a + b} + c}_{m n}$ Definition of Matrix Entrywise Addition $\ds$ $=$ $\ds \sqbrk {a + \paren {b + c} }_{m n}$ Associative Law of Addition $\ds$ $=$ $\ds \sqbrk a_{m n} + \sqbrk {b + c}_{m n}$ Definition of Matrix Entrywise Addition $\ds$ $=$ $\ds \sqbrk a_{m n} + \paren {\sqbrk b_{m n} + \sqbrk c_{m n} }$ Definition of Matrix Entrywise Addition $\ds$ $=$ $\ds \mathbf A + \paren {\mathbf B + \mathbf C}$ Definition of $\mathbf A$, $\mathbf B$ and $\mathbf C$

$\blacksquare$