Ring of Square Matrices over Real Numbers/Examples/2 x 2

== Example of Ring of Square Matrices over Real Numbers ==
Let $\struct {\map {\MM_\R} 2, +, \times}$ denote the ring of square matrices of order $2$ over the real numbers $\R$.
Then $\struct {\map {\MM_\R} 2, +, \times}$ forms a ring with unity which is specifically not commutative and also not an integral domain.
== Proof ==
We need to do the following:
:$(1): \quad$ Demonstrate that $\struct {\map {\MM_\R} 2, +, \times}$ satisfies the ring axioms
:$(2): \quad$ Demonstrate that matrix product on $\map {\MM_\R} 2$ is not commutative
:$(3): \quad$ Demonstrate that $\struct {\map {\MM_\R} 2, +, \times}$ has a unity
:$(4): \quad$ Demonstrate the existence of proper zero divisors for $\struct {\map {\MM_\R} 2, +, \times}$.
=== $(1)$: Ring Axioms ===
We check the ring axioms:
Let $\mathbf A = \sqbrk a_{i j}$, $\mathbf B = \sqbrk b_{i j}$ and $\mathbf C = \sqbrk c_{i j}$ where:
:$i, j \in \set {1, 2}$
:$a_{i j}, b_{i j}, c_{i j} \in \R$
be arbitrary real elements of $\map {\MM_\R} 2$.
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From Matrix Entrywise Addition: $2 \times 2$ Real Matrices, matrix addition on $\map {\MM_\R} 2$ is closed.
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Thus matrix addition is associative on $\map {\MM_\R} 2$
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Thus matrix addition is commutative on $\map {\MM_\R} 2$.
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We have:
Thus $\begin {pmatrix} 0 & 0 \\ 0 & 0 \end {pmatrix}$ is seen to be the identity element of matrix addition on $\map {\MM_\R} 2$.
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We have:
Thus $\begin {pmatrix} -a_{11} & -a_{12} \\ -a_{21} & -a_{22} \end {pmatrix}$ is seen to be the inverse element of $\begin {pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end {pmatrix}$ for matrix addition on $\map {\MM_\R} 2$.
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From Matrix Product (Conventional): $2 \times 2$ Real Matrices, matrix product on $\map {\MM_\R} 2$ is closed.
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From Real Addition is Commutative, the expression for $\mathbf A \paren {\mathbf B \mathbf C}$ is seen to equal the one for $\paren {\mathbf A \mathbf B} \mathbf C$.
Thus matrix product is associative on $\map {\MM_\R} 2$.
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The expression for $\mathbf A \paren {\mathbf B + \mathbf C}$ is seen to equal the one for $\mathbf A \mathbf B + \mathbf A \mathbf C$.
Similarly:
The expression for $\paren {\mathbf A + \mathbf B} \mathbf C$ is seen to equal the one for $\mathbf A \mathbf C + \mathbf B \mathbf C$.
Thus matrix product is distributive over matrix addition on $\map {\MM_\R} 2$.
=== $(2)$: Non-Commutativity of Matrix Product ===
Let:
By definition, both $\mathbf A$ and $\mathbf B$ are elements of $\map {\MM_\R} 2$.
We have:
and:
and it is seen that:
:$\mathbf A \mathbf B \ne \mathbf B \mathbf A$
Thus by definition matrix product is not commutative on $\map {\MM_\R} 2$.
=== $(3)$: Existence of Unity ===
We have:
and:
demonstrating that $\begin {pmatrix} 1 & 0 \\ 0 & 1 \end {pmatrix}$ serves as a unity for $\struct {\map {\MM_\R} 2, +, \times}$.
=== $(4)$: Existence of Proper Zero Divisors ===
As for the proof of non-commutativity of matrix product on $\struct {\map {\MM_\R} 2, +, \times}$, let:
Recall that we have:
But $\begin {pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$ is the zero of $\struct {\map {\MM_\R} 2, +, \times}$.
Thus $\struct {\map {\MM_\R} 2, +, \times}$ has proper zero divisors.
It follows that $\struct {\map {\MM_\R} 2, +, \times}$ is not an integral domain.
== Sources ==
* : Chapter $1$: Integral Domains: $\S 3$. Definition of an Integral Domain: Example $3$
* : Chapter $5$: Rings: $\S 18$. Definition of a Ring: Example $27$
Category:Ring of Square Matrices over Real Numbers