Find the inverse of the matrix \(A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 1 & 0 & 1 \end{bmatrix} \).

Step 1 ：First, let's find the determinant of matrix A: \(det(A) = 1(1(1) - 4(0)) - 2(0(1) - 4(1)) + 3(0(0) - 1(1)) = 1 - 8 - 3 = -10\).

Step 2 ：Now, let's find the adjoint of matrix A: \(adj(A) = \begin{bmatrix} 1 - 4 & -(0 - 4) & 0 - 2 \\ -(0 - 4) & 1(1) - 3(1) & -(1 - 0) \\ 1 - 0 & -(1 - 3) & 1(1) - 2(0) \end{bmatrix} = \begin{bmatrix} -3 & 4 & -2 \\ 4 & -2 & -1 \\ 1 & -2 & 1 \end{bmatrix}\).

Step 3 ：Finally, let's find the inverse of matrix A: \(A^{-1} = \frac{1}{det(A)} adj(A) = \frac{1}{-10} \begin{bmatrix} -3 & 4 & -2 \\ 4 & -2 & -1 \\ 1 & -2 & 1 \end{bmatrix} = \begin{bmatrix} 0.3 & -0.4 & 0.2 \\ -0.4 & 0.2 & 0.1 \\ -0.1 & 0.2 & -0.1 \end{bmatrix}\).