The matrix is the "real" definition of the transformation. The angles and shifts are just used to generate the matrix. 3dAllineate does the search for parameters in angle+shift coordinates, and at each set of angle+shift parameters, generates the matrix to use for the coordinate transformation being evaluated.

One problem with going between angle+shift parameters and the matrix is that the matrix depends on the order in which the rotations and shifts are applied. The order of rotations is fixed in the program as R = Y X Z where Z is rotation about the I-S axis, X is rotation about the R-L axis, and Y is rotation about the A-P axis. The order of shift is not fixed. By default it is 'after' the rotation, so that the final matrix is T = S Y X Z (each matrix is a 4x4 transformation, with the last row being [0 0 0 1], of course). However, the inverse of such a matrix is Tinv = Zinv Xinv Yinv Sinv which isn't in the same order. That is, computing the inverse parameters from the inverse matrix isn't trivial.

At this point, I'm finished with random lecture mode, and hoping this helped. I'm not sure what more to say since I'm not sure what the specific question is.