Square Order 2 Matrices over Real Numbers form Ring with Unity

Theorem

Let $S$ denote the set of square matrices of order $2$ whose entries are the set of real numbers.

Then $S$ forms a non-commutative ring with unity whose unity is the matrix $\begin {pmatrix} 1 & 0 \\ 0 & 1 \end {pmatrix}$.

Proof

This is an instance of Ring of Square Matrices over Field is Ring with Unity.

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 55 \ (2)$. Special types of ring and ring elements
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): ring
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): ring