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

Example of Ring of Square Matrices over Real Numbers
Let $\struct {\map {\mathcal M_\R} 2, +, \times}$ denote the ring of square matrices of order $2$ over the real numbers $\R$.

Then $\struct {\map {\mathcal M_\R} 2, +, \times}$ 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 $\struct {\map {\mathcal M_\R} 2, +, \times}$ satisfies the ring axioms


 * $(2): \quad$ Demonstrate that matrix product on $\map {\mathcal M_\R} 2$ is not commutative


 * $(3): \quad$ Demonstrate that $\struct {\map {\mathcal M_\R} 2, +, \times}$ has a unity


 * $(4): \quad$ Demonstrate the existence of proper zero divisors for $\struct {\map {\mathcal M_\R} 2, +, \times}$.

$(1)$: 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 $\map {\mathcal M_\R} 2$.

$A0$: Closure under Addition
From Matrix Entrywise Addition: $2 \times 2$ Real Matrices, matrix addition on $\map {\mathcal M_\R} 2$ is closed.

$A1$: Associativity of Addition
Thus matrix addition is associative on $\map {\mathcal M_\R} 2$

$A2$: Commutativity of Addition
Thus matrix addition is commutative on $\map {\mathcal M_\R} 2$.

$A3$: Identity Element for Addition
We have:

Thus $\begin {pmatrix} 0 & 0 \\ 0 & 0 \end {pmatrix}$ is seen to be the identity element of matrix addition on $\map {\mathcal M_\R} 2$.

$A4$: Inverse Elements for Addition
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 $\map {\mathcal M_\R} 2$.

$M0$: Closure under Multiplication
From Matrix Product (Conventional): $2 \times 2$ Real Matrices, matrix product on $\map {\mathcal M_\R} 2$ is closed.

$M1$: Associativity of Multiplication
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 $\map {\mathcal M_\R} 2$.

$D$: Distributivity of Multiplication over Addition
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 $\map {\mathcal M_\R} 2$.

$(2)$: Non-Commutativity of Matrix Product
Let:

By definition, both $\mathbf A$ and $\mathbf B$ are elements of $\map {\mathcal M_\R} 2$.

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 $\map {\mathcal M_\R} 2$.

$(3)$: Existence of Unity
We have:

and:

demonstrating that $\begin {pmatrix} 1 & 0 \\ 0 & 1 \end {pmatrix}$ serves as a unity for $\struct {\map {\mathcal M_\R} 2, +, \times}$.

$(4)$: Existence of Proper Zero Divisors
As for the proof of non-commutativity of matrix product on $\struct {\map {\mathcal M_\R} 2, +, \times}$, let:

Recall that we have:

But $\begin {pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$ is the zero of $\struct {\map {\mathcal M_\R} 2, +, \times}$.

Thus $\struct {\map {\mathcal M_\R} 2, +, \times}$ has proper zero divisors.

It follows that $\struct {\map {\mathcal M_\R} 2, +, \times}$ is not an integral domain.