Identify the given matrix and the values of its elements.

The matrix $A$ is given as:

$$A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$$

The given values for the elements are:
$a_{11} = 1$
$a_{22} = 1$
$a_{21} = 2$
$a_{12} = 2$

For a $2 \times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the determinant is calculated as $ad - bc$.
Applying this formula to matrix $A$:

$$|A| = (1)(1) - (2)(2)$$ $$|A| = 1 - 4$$ $$|A| = -3$$

The calculated determinant is $-3$. This matches option D.

The final answer is $\text{D. -3}$.