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 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.