Cofactor matrix 4x4, evaluated by hand

I am currently studying linear algebra. So far I have studied method for aquiring the inverse of a matrix A. Now I would like to evaluate the inverse of a $4\times 4$ matrix using the following formula:

$$\mathbf{A}^{-1} = \frac{ \mathrm{adj}(\mathbf{A}) }{ \det(\mathbf{A}) } % A^(-1)=1/det⁡(A) adj(A)$$

My question is in order to get the adjoint matrix I will need the cofactor matrix. In a $4\times 4$ matrix this means I could do it with row expansion in 2 dimensions. For instance, I take the entry A[a11], I cross out first row and first column. Next I choose B[a11] in the submatrix to get my minor. Now, do I have to multiply the minor of the submatrix with the original entry A[a11]? Does it matter what minor I choose in the submatrix in further calculation? For example, using A[a12] to get the minor of the main matrix, can I use any row or column in the submatrix?

I appreciate any help in this manner. Thank you.

-Daniel

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That was quite incomprehensible... But you know how to compute a 3 by 3 determinant, right? That's what you have to do (16 times!), since each cofactor is a 3 by 3 determinant (namely the determinant that you get by crossing out one row and one column of $A$). – Hans Lundmark Oct 11 '12 at 7:48