Although the three necessary elementary parameters have been defined, at this point the two-dimensional rigid-body transformation is still ambiguous. This is due to the fact that the order of the three elementary operations has not been specified. Two reasonable orderings are:

This matrix can be decomposed back into the original sequence of elementary transformations, or alternatively can be decomposed into the reverse sequence. If translations are applied before rotations, the magnitude and direction of the rotation around the origin is unchanged, but it must be preceded by a translation of 2.3764 units along x and 9.5579 units along y:

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