Finding the Adjugate of a 4x4 Matrix

[Philosophaie]

I need to find the adj(A) for a 4x4 Matrix.

$$A = \begin{array} a11 & a12 & a13 & a14 \\ a21 & a22 & a23 & a24 \\ a31 & a32 & a33 & a34 \\ a41 & a42 & a43 & a44 \end{array}$$

I have tried:

$$adj(A_{ij}) = (-1)^{i+j}*A_{ji}$$

but I get the wrong answer for the inverse:

$$A^{-1} = \frac{adj(A)}{det(A)}$$

and this does not work:

$$A*A^{-1} = I$$

where

$$I= \begin{array} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}$$

 

[Simon Bridge]

$C_{ij}=(-1)^{i+j}A_{ij}$ ... gives you $\text{C}$ - the cofactor matrix.
The adjugate matrix is the transpose of the cofactor matrix.