I know about the Kramer's rule (and have that working), the Gaussian method, and know that the inverse of a rotational 3x3 matrix is it's transpose. But I think I'm missing something. Since the matrix is orthogonal and I can calculate the 3x3 part of it by transposing it, shouldn't there be some easier method to calculate the remaining 3 numbers (the translate part)? I heard somewhere that they should be flipped from positive to negative, but that's not quite working. What am I missing?

Here's an example... I have this matrix:

0  4  8  12
1  5  9  13
2  6  10 14
3  7  11 15

X:
0.819300
-0.196000
0.538800
-0.591540
Y:
0.301200
0.946800
-0.113700
-1.476260
Z:
-0.487900
0.255500
0.834700
-0.163410
W:
0.000000
0.000000
0.000000
1.000000


Using the Kramer's rule, the inverse of it is:
X:
0.819302
0.301249
-0.487825
0.849656
Y:
-0.195927
0.946701
0.255428
1.323417
Z:
0.538872
-0.113697
0.834705
0.287318
W:
0.000000
0.000000
0.000000
1.000000


But the center can be found by simply transposing the 3x3 part:
X:
0.819300…

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