Find the determinant of the following 4x4 matrix: \[A = \begin{pmatrix} 1 & 2 & 3 & 4 \\ 4 & 3 & 2 & 1 \\ 1 & 1 & 1 & 1 \\ 2 & 2 & 2 & 2 \end{pmatrix}\]

Step 1 ：Step 1: Apply the determinant properties and expand the determinant using the first column of the matrix. This gives us: \[\det(A) = 1\cdot\det\begin{pmatrix} 3 & 2 & 1 \\ 1 & 1 & 1 \\ 2 & 2 & 2 \end{pmatrix} - 4\cdot\det\begin{pmatrix} 2 & 3 & 4 \\ 1 & 1 & 1 \\ 2 & 2 & 2 \end{pmatrix} + 1\cdot\det\begin{pmatrix} 2 & 3 & 4 \\ 4 & 2 & 1 \\ 2 & 2 & 2 \end{pmatrix} - 2\cdot\det\begin{pmatrix} 2 & 3 & 4 \\ 4 & 2 & 1 \\ 1 & 1 & 1 \end{pmatrix}\]

Step 2 ：Step 2: Compute each 3x3 determinant as in step 1. This gives us: \[\det(A) = 1\cdot(-4) - 4\cdot(-6) + 1\cdot(-6) - 2\cdot(-4) = -4 + 24 - 6 + 8 = 22\]