Calculate the matrix multiplication Ax for the given matrices.

Step 1: Calculate the first element of the resulting matrix, $c_{1,1}$. This is obtained by multiplying the elements of the first row of the first matrix by the corresponding elements of the column vector and summing them. $c_{1,1} = (6)(4) + (2)(-2) + (4)(1)$ $c_{1,1} = 24 - 4 + 4$ $c_{1,1} = 24$

Step 2: Calculate the second element of the resulting matrix, $c_{2,1}$. This is obtained by multiplying the elements of the second row of the first matrix by the corresponding elements of the column vector and summing them. $c_{2,1} = (-1)(4) + (4)(-2) + (3)(1)$ $c_{2,1} = -4 - 8 + 3$ $c_{2,1} = -12 + 3$ $c_{2,1} = -9$

Step 3: Calculate the third element of the resulting matrix, $c_{3,1}$. This is obtained by multiplying the elements of the third row of the first matrix by the corresponding elements of the column vector and summing them. $c_{3,1} = (-2)(4) + (9)(-2) + (3)(1)$ $c_{3,1} = -8 - 18 + 3$ $c_{3,1} = -26 + 3$ $c_{3,1} = -23$

Step 4: Assemble the resulting matrix. $Ax = \begin{bmatrix} c_{1,1} \\ c_{2,1} \\ c_{3,1} \end{bmatrix} = \begin{bmatrix} 24 \\ -9 \\ -23 \end{bmatrix}$

The final answer is $\boxed{\begin{bmatrix} 24 \\ -9 \\ -23 \end{bmatrix}}$.