Step 1: 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: $a_{11} = 1$ $a_{22} = 1$ $a_{21} = 2$ $a_{12} = 2$ Step 2: Substitute the given values into the matrix $A$. $$A = \begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix}$$ Step 3: Calculate the determinant of $A$, denoted as $|A|$. 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 $A$: $$|A| = (1)(1) - (2)(2)$$ $$|A| = 1 - 4$$ $$|A| = -3$$ Step 4: Compare the result with the given options. The calculated determinant is $-3$. This matches option D. The final answer is $\boxed{\text{D. -3}}$.