1. The problem asks to find the inverse of the matrix $$\begin{bmatrix}1 & 2 \\ 2 & 4\end{bmatrix}$$. 2. The formula for the inverse of a 2x2 matrix $$A = \begin{bmatrix}a & b \\ c & d\end{bmatrix}$$ is: $$A^{-1} = \frac{1}{ad - bc} \begin{bmatrix}d & -b \\ -c & a\end{bmatrix}$$ where $$ad - bc$$ is the determinant of the matrix. 3. Calculate the determinant of the given matrix: $$\det = (1)(4) - (2)(2) = 4 - 4 = 0$$ 4. Since the determinant is zero, the matrix is singular and does not have an inverse. 5. Therefore, the correct answer is that the inverse does not exist.