The combination of 2 symmetry operations follows the procedure of Section 2.2. In analogy to equations (2.2.5) to (2.2.8) one obtains

These equations may be formulated with matrix-column pairs:

Note that in the product (

By comparing equations (4.2.4) and (4.2.5) one obtains

This

The multiplication of matrix-column pairs is *associative*, because

(4.2.7) |

and on the other hand,

(4.2.8) |

By comparison of both expressions one finds

Associativity is a very important property. It can be used,

A linear mapping is a mapping which leaves the origin fixed. Its column
part is thus the * o* column. According to equation (4.2.6)
any matrix-column pair can be decomposed into a linear mapping (

Question: What is the result if the translation
(* I*,

Before the reversion of a symmetry operation is dealt with, a general
remark is appropriate. In general, the formulae of this section are not
restricted to crystallographic symmetry operations but are valid also
for affine mappings. However, there is one exception. In the inversion
of a matrix * W* the determinant
appears in the
denominator of the coefficients of
, see Subsection
2.6.1. Therefore, the condition
has to be
fulfilled. Such mappings are called

Now to the calculation of the reverse of a matrix-column pair. It is
often necessary to know which matrix * C* and column

This equation is as unpleasant as is equation (4.2.6). The matrix part is fine but the column part is not just as one would like to see but has to be multiplied with . The next section will present a proposal how to overcome this inconvenience.

It is always good to test the result of a calculation or derivation. One verifies the validity of the equations by applying equations (4.2.6) and (4.2.12). In addition in the following Problem 2A the results of this section may be practised.

Problem 2A. Symmetry described by matrix-column pairs.

In Vol. A of *International Tables for Crystallography* the
crystallographic symmetry operations *A*, *B*, ... are referred
to a conventional coordinate system and are represented by matrix-column
pairs (* A*,

and

Combining two symmetry operations or reversion of a symmetry operation corresponds to multiplication or reversion of these matrix-column pairs, such that the resulting matrix-column pair represents the resulting symmetry operation.

The following calculations make use of the formulae 4.2.6 and
4.2.12.

Can one exploit the fact that the matrices * A*,

Questions

**(i)**- What is the matrix-column pair resulting from

(,**B**)(**b**,**A**) = (**a**,**C**) ?**c** **(ii)**- What is the matrix-column pair resulting from

(,**A**)(**a**,**B**) = (**b**,**D**) ?**d** **(iii)**- What is (
,**A**) ?**a** **(iv)**- What is (
,**B**) ?**b** **(v)**- What is (
,**C**) ?**c** **(vi)**- What is (
,**D**) ?**d** **(vii)**- What is (
,**B**)(**b**,**A**) ?**a**

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