Matrix Entrywise Addition over Ring is Associative/Proof 1
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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
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 $\mathbf C$ |
$\blacksquare$