1. The problem is 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 $$\begin{bmatrix}a & b \\ c & d\end{bmatrix}$$ is $$\frac{1}{ad - bc} \begin{bmatrix}d & -b \\ -c & a\end{bmatrix}$$, provided that the determinant $$ad - bc \neq 0$$. 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 inverse of the matrix does not exist because the determinant is zero, which means the matrix is not invertible.