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

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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:

A ring is an algebraic structure $\struct {R, *, \circ}$, on which are defined two binary operations $\circ$ and $*$, which satisfy the following conditions:

\((\text A 0)\)   $:$   Closure under addition      \(\ds \forall a, b \in R:\) \(\ds a * b \in R \)             
\((\text A 1)\)   $:$   Associativity of addition      \(\ds \forall a, b, c \in R:\) \(\ds \paren {a * b} * c = a * \paren {b * c} \)             
\((\text A 2)\)   $:$   Commutativity of addition      \(\ds \forall a, b \in R:\) \(\ds a * b = b * a \)             
\((\text A 3)\)   $:$   Identity element for addition: the zero      \(\ds \exists 0_R \in R: \forall a \in R:\) \(\ds a * 0_R = a = 0_R * a \)             
\((\text A 4)\)   $:$   Inverse elements for addition: negative elements      \(\ds \forall a \in R: \exists a' \in R:\) \(\ds a * a' = 0_R = a' * a \)             
\((\text M 0)\)   $:$   Closure under product      \(\ds \forall a, b \in R:\) \(\ds a \circ b \in R \)             
\((\text M 1)\)   $:$   Associativity of product      \(\ds \forall a, b, c \in R:\) \(\ds \paren {a \circ b} \circ c = a \circ \paren {b \circ c} \)             
\((\text D)\)   $:$   Product is distributive over addition      \(\ds \forall a, b, c \in R:\) \(\ds a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c} \)             
\(\ds \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c} \)             

These criteria are called 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 $\map {\MM_\R} 2$.


Ring Axiom $\text A0$: Closure under Addition

From Matrix Entrywise Addition: $2 \times 2$ Real Matrices, matrix addition on $\map {\MM_\R} 2$ is closed.

$\Box$


Ring Axiom $\text A1$: Associativity of Addition

\(\ds \mathbf A + \paren {\mathbf B + \mathbf C}\) \(=\) \(\ds \sqbrk a_{i j} + \paren {\sqbrk b_{i j} + \sqbrk c_{i j} }\)
\(\ds \) \(=\) \(\ds \sqbrk a_{i j} + \sqbrk {b + c}_{i j}\) Definition of Matrix Entrywise Addition
\(\ds \) \(=\) \(\ds \sqbrk {a + \paren {b + c} }_{i j}\) Definition of Matrix Entrywise Addition
\(\ds \) \(=\) \(\ds \sqbrk {\paren {a + b} + c}_{i j}\) Real Addition is Associative
\(\ds \) \(=\) \(\ds \sqbrk {a + b}_{i j} + \sqbrk c_{i j}\) Definition of Matrix Entrywise Addition
\(\ds \) \(=\) \(\ds \paren {\sqbrk a_{i j} + \sqbrk b_{i j} } + \sqbrk c_{i j}\) Definition of Matrix Entrywise Addition
\(\ds \) \(=\) \(\ds \paren {\mathbf A + \mathbf B} + \mathbf C\)

Thus matrix addition is associative on $\map {\MM_\R} 2$

$\Box$


Ring Axiom $\text A2$: Commutativity of Addition

\(\ds \mathbf A + \mathbf B\) \(=\) \(\ds \sqbrk a_{i j} + \sqbrk b_{i j}\)
\(\ds \) \(=\) \(\ds \sqbrk {a + b}_{i j}\) Definition of Matrix Entrywise Addition
\(\ds \) \(=\) \(\ds \sqbrk {b + a}_{i j}\) Real Addition is Commutative
\(\ds \) \(=\) \(\ds \mathbf B + \mathbf A\)

Thus matrix addition is commutative on $\map {\MM_\R} 2$.

$\Box$


Ring Axiom $\text A3$: Identity for Addition

We have:

\(\ds \begin {pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end {pmatrix} + \begin {pmatrix} 0 & 0 \\ 0 & 0 \end {pmatrix}\) \(=\) \(\ds \begin {pmatrix} a_{11} + 0 & a_{12} + 0 \\ a_{21} + 0 & a_{22} + 0 \end {pmatrix}\) Definition of Matrix Entrywise Addition
\(\ds \) \(=\) \(\ds \begin {pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end {pmatrix}\) Real Addition Identity is Zero
\(\ds \) \(=\) \(\ds \begin {pmatrix} 0 + a_{11} & 0 + a_{12} \\ 0 + a_{21} & 0 + a_{22} \end {pmatrix}\) Real Addition is Commutative
\(\ds \) \(=\) \(\ds \begin {pmatrix} 0 & 0 \\ 0 & 0 \end {pmatrix} + \begin {pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end {pmatrix}\)

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

$\Box$


Ring Axiom $\text A4$: Inverses for Addition

We have:

\(\ds \begin {pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end {pmatrix} + \begin {pmatrix} -a_{11} & -a_{12} \\ -a_{21} & -a_{22} \end {pmatrix}\) \(=\) \(\ds \begin {pmatrix} a_{11} + \paren {-a_{11} } & a_{12} + \paren {-a_{12} } \\ a_{21} + \paren {-a_{21} } & a_{22} + \paren {-a_{22} } \end {pmatrix}\) Definition of Matrix Entrywise Addition
\(\ds \) \(=\) \(\ds \begin {pmatrix} 0 & 0 \\ 0 & 0 \end {pmatrix}\)
\(\ds \) \(=\) \(\ds \begin {pmatrix} \paren {-a_{11} } + a_{11} & \paren {-a_{12} } + a_{12} \\ \paren {-a_{21} } + a_{21} & \paren {-a_{22} } + a_{22} \end {pmatrix}\)
\(\ds \) \(=\) \(\ds \begin {pmatrix} -a_{11} & -a_{12} \\ -a_{21} & -a_{22} \end {pmatrix} + \begin {pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end {pmatrix}\)

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 $\map {\MM_\R} 2$.

$\Box$


Ring Axiom $\text M0$: Closure under Product

From Matrix Product (Conventional): $2 \times 2$ Real Matrices, matrix product on $\map {\MM_\R} 2$ is closed.

$\Box$


Ring Axiom $\text M1$: Associativity of Product

\(\ds \mathbf A \paren {\mathbf B \mathbf C}\) \(=\) \(\ds \begin {pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end {pmatrix} \paren {\begin {pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end {pmatrix} \begin {pmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end {pmatrix} }\)
\(\ds \) \(=\) \(\ds \begin {pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end {pmatrix} \begin {pmatrix} b_{11} c_{11} + b_{12} c_{21} & b_{11} c_{12} + b_{12} c_{22} \\ b_{21} c_{11} + b_{22} c_{21} & b_{21} c_{12} + b_{22} c_{22} \end {pmatrix}\) Definition of Matrix Product (Conventional)
\(\ds \) \(=\) \(\ds \begin {pmatrix} a_{11} \paren {b_{11} c_{11} + b_{12} c_{21} } + a_{12} \paren {b_{21} c_{11} + b_{22} c_{21} } & a_{11} \paren {b_{11} c_{12} + b_{12} c_{22} } + a_{12} \paren {b_{21} c_{12} + b_{22} c_{22} } \\ a_{21} \paren {b_{11} c_{11} + b_{12} c_{21} } + a_{22} \paren {b_{21} c_{11} + b_{22} c_{21} } & a_{21} \paren {b_{11} c_{12} + b_{12} c_{22} } + a_{22} \paren {b_{21} c_{12} + b_{22} c_{22} } \end {pmatrix}\) Definition of Matrix Product (Conventional)
\(\ds \) \(=\) \(\ds \begin {pmatrix} a_{11} b_{11} c_{11} + a_{11} b_{12} c_{21} + a_{12} b_{21} c_{11} + a_{12} b_{22} c_{21} & a_{11} b_{11} c_{12} + a_{11} b_{12} c_{22} + a_{12} b_{21} c_{12} + a_{12} b_{22} c_{22} \\ a_{21} b_{11} c_{11} + a_{21} b_{12} c_{21} + a_{22} b_{21} c_{11} + a_{22} b_{22} c_{21} & a_{21} b_{11} c_{12} + a_{21} b_{12} c_{22} + a_{22} b_{21} c_{12} + a_{22} b_{22} c_{22} \end {pmatrix}\) Real Multiplication Distributes over Addition


\(\ds \paren {\mathbf A \mathbf B} \mathbf C\) \(=\) \(\ds \paren {\begin {pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end {pmatrix} \begin {pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end {pmatrix} } \begin {pmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end {pmatrix}\)
\(\ds \) \(=\) \(\ds \begin {pmatrix} a_{11} b_{11} + a_{12} b_{21} & a_{11} b_{12} + a_{12} b_{22} \\ a_{21} b_{11} + a_{22} b_{21} & a_{21} b_{12} + a_{22} b_{22} \end {pmatrix} \begin {pmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end {pmatrix}\) Definition of Matrix Product (Conventional)
\(\ds \) \(=\) \(\ds \begin {pmatrix} \paren {a_{11} b_{11} + a_{12} b_{21} } c_{11} + \paren {a_{11} b_{12} + a_{12} b_{22} } c_{21} & \paren {a_{11} b_{11} + a_{12} b_{21} } c_{12} + \paren {a_{11} b_{12} + a_{12} b_{22} } c_{22} \\ \paren {a_{21} b_{11} + a_{22} b_{21} } c_{11} + \paren {a_{21} b_{12} + a_{22} b_{22} } c_{21} & \paren {a_{21} b_{11} + a_{22} b_{21} } c_{12} + \paren {a_{21} b_{12} + a_{22} b_{22} } c_{22} \end {pmatrix}\) Definition of Matrix Product (Conventional)
\(\ds \) \(=\) \(\ds \begin {pmatrix} a_{11} b_{11} c_{11} + a_{12} b_{21} c_{11} + a_{11} b_{12} c_{21} + a_{12} b_{22} c_{21} & a_{11} b_{11} c_{12} + a_{12} b_{21} c_{12} + a_{11} b_{12} c_{22} + a_{12} b_{22} c_{22} \\ a_{21} b_{11} c_{11} + a_{22} b_{21} c_{11} + a_{21} b_{12} c_{21} + a_{22} b_{22} c_{21} & a_{21} b_{11} c_{12} + a_{22} b_{21} c_{12} + a_{21} b_{12} c_{22} + a_{22} b_{22} c_{22} \end {pmatrix}\) Real Multiplication Distributes over Addition


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 $\map {\MM_\R} 2$.

$\Box$


Ring Axiom $\text D$: Distributivity of Product over Addition

\(\ds \mathbf A \paren {\mathbf B + \mathbf C}\) \(=\) \(\ds \begin {pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end {pmatrix} \paren {\begin {pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end {pmatrix} + \begin {pmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end {pmatrix} }\)
\(\ds \) \(=\) \(\ds \begin {pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end {pmatrix} \begin {pmatrix} b_{11} + c_{11} & b_{12} + c_{12} \\ b_{21} + c_{21} & b_{22} + c_{22} \end {pmatrix}\) Definition of Matrix Entrywise Addition
\(\ds \) \(=\) \(\ds \begin {pmatrix} a_{11} \paren {b_{11} + c_{11} } + a_{12} \paren {b_{21} + c_{21} } & a_{11} \paren {b_{12} + c_{12} } + a_{12} \paren {b_{22} + c_{22} } \\ a_{21} \paren {b_{11} + c_{11} } + a_{22} \paren {b_{21} + c_{21} } & a_{21} \paren {b_{12} + c_{12} } + a_{22} \paren {b_{22} + c_{22} } \end {pmatrix}\) Definition of Matrix Product (Conventional)
\(\ds \) \(=\) \(\ds \begin {pmatrix} a_{11} b_{11} + a_{11} c_{11} + a_{12} b_{21} + a_{12} c_{21} & a_{11} b_{12} + a_{11} c_{12} + a_{12} b_{22} + a_{12} c_{22} \\ a_{21} b_{11} + a_{21} c_{11} + a_{22} b_{21} + a_{22} c_{21} & a_{21} b_{12} + a_{21} c_{12} + a_{22} b_{22} + a_{22} c_{22} \end {pmatrix}\) Real Multiplication Distributes over Addition


\(\ds \mathbf A \mathbf B + \mathbf A \mathbf C\) \(=\) \(\ds \begin {pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end {pmatrix} \begin {pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end {pmatrix} + \begin {pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end {pmatrix} \begin {pmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end {pmatrix}\)
\(\ds \) \(=\) \(\ds \begin {pmatrix} a_{11} b_{11} + a_{12} b_{21} & a_{11} b_{12} + a_{12} b_{22} \\ a_{21} b_{11} + a_{22} b_{21} & a_{21} b_{12} + a_{22} b_{22} \end {pmatrix} + \begin {pmatrix} a_{11} c_{11} + a_{12} c_{21} & a_{11} c_{12} + a_{12} c_{22} \\ a_{21} c_{11} + a_{22} c_{21} & a_{21} c_{12} + a_{22} c_{22} \end {pmatrix}\) Definition of Matrix Product (Conventional)
\(\ds \) \(=\) \(\ds \begin {pmatrix} a_{11} b_{11} + a_{11} c_{11} + a_{12} b_{21} + a_{12} c_{21} & a_{11} b_{12} + a_{11} c_{12} + a_{12} b_{22} + a_{12} c_{22} \\ a_{21} b_{11} + a_{21} c_{11} + a_{22} b_{21} + a_{22} c_{21} & a_{21} b_{12} + a_{21} c_{12} + a_{22} b_{22} + a_{22} c_{22} \end {pmatrix}\) Definition of Matrix Entrywise 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:

\(\ds \paren {\mathbf A + \mathbf B} \mathbf C\) \(=\) \(\ds \paren {\begin {pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end {pmatrix} + \begin {pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end {pmatrix} } \begin {pmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end {pmatrix}\)
\(\ds \) \(=\) \(\ds \begin {pmatrix} a_{11} + b_{11} & a_{12} + b_{12} \\ a_{21} + b_{21} & a_{22} + b_{22} \end {pmatrix} \begin {pmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end {pmatrix}\) Definition of Matrix Entrywise Addition
\(\ds \) \(=\) \(\ds \begin {pmatrix} \paren {a_{11} + b_{11} } c_{11} + \paren {a_{12} + b_{12} } c_{21} & \paren {a_{11} + b_{11} } c_{12} + \paren {a_{12} + b_{12} } c_{22} \\ \paren {a_{21} + b_{21} } c_{11} + \paren {a_{22} + b_{22} } c_{21} & \paren {a_{21} + b_{21} } c_{12} + \paren {a_{22} + b_{22} } c_{22} \end {pmatrix}\) Definition of Matrix Product (Conventional)
\(\ds \) \(=\) \(\ds \begin {pmatrix} a_{11} c_{11} + b_{11} c_{11} + a_{12} c_{21} + b_{12} c_{21} & a_{11} c_{12} + b_{11} c_{12} + a_{12} c_{22} + b_{12} c_{22} \\ a_{21} c_{11} + b_{21} c_{11} + a_{22} c_{21} + b_{22} c_{21} & a_{21} c_{12} + b_{21} c_{12} + a_{22} c_{22} + b_{22} c_{22} \end {pmatrix}\) Real Multiplication Distributes over Addition


\(\ds \mathbf A \mathbf C + \mathbf B \mathbf C\) \(=\) \(\ds \begin {pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end {pmatrix} \begin {pmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end {pmatrix} + \begin {pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end {pmatrix} \begin {pmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end {pmatrix}\)
\(\ds \) \(=\) \(\ds \begin {pmatrix} a_{11} c_{11} + a_{12} c_{21} & a_{11} c_{12} + a_{12} c_{22} \\ a_{21} c_{11} + a_{22} c_{21} & a_{21} c_{12} + a_{22} c_{22} \end {pmatrix} + \begin {pmatrix} b_{11} c_{11} + b_{12} c_{21} & b_{11} c_{12} + b_{12} c_{22} \\ b_{21} c_{11} + b_{22} c_{21} & b_{21} c_{12} + b_{22} c_{22} \end {pmatrix}\) Definition of Matrix Product (Conventional)
\(\ds \) \(=\) \(\ds \begin {pmatrix} a_{11} c_{11} + b_{11} c_{11} + a_{12} c_{21} + b_{12} c_{21} & a_{11} c_{12} + b_{11} c_{12} + a_{12} c_{22} + b_{12} c_{22} \\ a_{21} c_{11} + b_{21} c_{11} + a_{22} c_{21} + b_{22} c_{21} & a_{21} c_{12} + b_{21} c_{12} + a_{22} c_{22} + b_{22} c_{22} \end {pmatrix}\) Definition of Matrix Entrywise Addition


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 $\map {\MM_\R} 2$.

$\Box$


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

Let:

\(\ds \mathbf A\) \(=\) \(\ds \begin {pmatrix} 0 & 1 \\ 0 & 0 \end {pmatrix}\)
\(\ds \mathbf B\) \(=\) \(\ds \begin {pmatrix} 1 & 0 \\ 0 & 0 \end {pmatrix}\)

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


We have:

\(\ds \mathbf A \mathbf B\) \(=\) \(\ds \begin {pmatrix} 0 & 1 \\ 0 & 0 \end {pmatrix} \begin {pmatrix} 1 & 0 \\ 0 & 0 \end {pmatrix}\)
\(\ds \) \(=\) \(\ds \begin {pmatrix} 0 \times 1 + 1 \times 0 & 0 \times 0 + 1 \times 0 \\ 0 \times 1 + 0 \times 0 & 0 \times 0 + 0 \times 0 \end{pmatrix}\)
\(\ds \) \(=\) \(\ds \begin {pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}\)


and:

\(\ds \mathbf B \mathbf A\) \(=\) \(\ds \begin {pmatrix} 1 & 0 \\ 0 & 0 \end {pmatrix} \begin {pmatrix} 0 & 1 \\ 0 & 0 \end {pmatrix}\)
\(\ds \) \(=\) \(\ds \begin {pmatrix} 1 \times 0 + 0 \times 0 & 1 \times 1 + 0 \times 0 \\ 0 \times 0 + 0 \times 0 & 0 \times 1 + 0 \times 0 \end{pmatrix}\)
\(\ds \) \(=\) \(\ds \begin {pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}\)


and it is seen that:

$\mathbf A \mathbf B \ne \mathbf B \mathbf A$

Thus by definition matrix product is not commutative on $\map {\MM_\R} 2$.

$\Box$


$(3)$: Existence of Unity

We have:

\(\ds \) \(\) \(\ds \begin {pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end {pmatrix} \begin {pmatrix} 1 & 0 \\ 0 & 1 \end {pmatrix}\)
\(\ds \) \(=\) \(\ds \begin {pmatrix} a_{11} \times 1 + a_{12} \times 0 & a_{11} \times 0 + a_{12} \times 1 \\ a_{21} \times 1 + a_{22} \times 0 & a_{21} \times 0 + a_{22} \times 1 \end {pmatrix}\) Definition of Matrix Product (Conventional)
\(\ds \) \(=\) \(\ds \begin {pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end {pmatrix}\)

and:

\(\ds \) \(\) \(\ds \begin {pmatrix} 1 & 0 \\ 0 & 1 \end {pmatrix} \begin {pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end {pmatrix}\)
\(\ds \) \(=\) \(\ds \begin {pmatrix} 1 \times a_{11} + 0 \times a_{21} & 1 \times a_{12} + 0 \times a_{22} \\ 0 \times a_{11} + 1 \times a_{21} & 0 \times a_{12} + 1 \times a_{22} \end {pmatrix}\) Definition of Matrix Product (Conventional)
\(\ds \) \(=\) \(\ds \begin {pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end {pmatrix}\)

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

$\Box$


$(4)$: Existence of Proper Zero Divisors

As for the proof of non-commutativity of matrix product on $\struct {\map {\MM_\R} 2, +, \times}$, let:

\(\ds \mathbf A\) \(=\) \(\ds \begin {pmatrix} 0 & 1 \\ 0 & 0 \end {pmatrix}\)
\(\ds \mathbf B\) \(=\) \(\ds \begin {pmatrix} 1 & 0 \\ 0 & 0 \end {pmatrix}\)


Recall that we have:

\(\ds \mathbf A \mathbf B\) \(=\) \(\ds \begin {pmatrix} 0 & 1 \\ 0 & 0 \end {pmatrix} \begin {pmatrix} 1 & 0 \\ 0 & 0 \end {pmatrix}\)
\(\ds \) \(=\) \(\ds \begin {pmatrix} 0 \times 1 + 1 \times 0 & 0 \times 0 + 1 \times 0 \\ 0 \times 1 + 0 \times 0 & 0 \times 0 + 0 \times 0 \end{pmatrix}\)
\(\ds \) \(=\) \(\ds \begin {pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}\)


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.

$\blacksquare$


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