Only a few applications can be dealt with here:

- The inversion of a matrix;
- Another formula for calculating the distance between points or the angle between lines (bindings);
- A formula for the volume of the unit cell of a crystal.

The inversion of a square matrix * A* is a task which occurs everywhere
in matrix calculations. Here we restrict the considerations to the inversion
of (2 2) and (3 3) matrices. In Least-Squares refiments the
inversion of huge matrices was a serious problem before the computers and
programs were sufficiently developed.

**Definition** (D 2.6.2) A matrix * C* which fulfills the condition
for a given matrix

The matrix * A* exists if and only if
.
In the following we assume

**Definition** (D 2.6.2) A matrix * A* is called

The name comes from the fact that the matrix part of any isometry is an orthogonal matrix if referred to an orthonormal basis. In crystallography most matrices of the crystallographic symmetry operations are orthogonal if referred to the conventional basis.

Procedure: One forms the transposed matrix * A*
from the given
matrix

There are several general methods to invert a matrix. Here we use a formula based on determinants. It is not restricted to dimensions 2 or 3.

Let * A* = be the matrix to be inverted,
its determinant, and

**Note** that in this equation the indices of
are **exchanged** with respect to the element
which is to be determined.

Example. Calculate the inverse matrix of

One determines
and
obtains for the coefficients of **A**

With these coefficients one finds

and verifies that holds.

Distances and angles

*fundamental matrix of the coordinate basis*

Because of , * G* is a symmetric matrix.

In the formulae of Section 1.6 one may replace the `index formalism'
by the `matrix formalism'. Using matrix multiplication with rows and columns,

This is the same as equation (1.6.3) but expressed in another way. Such
`matrix formulae' are useful in general calculations when changing the basis,
when describing the relation between crystal lattice and reciprocal lattice,
*etc.* However, for the actual calculation of distances, angles,
*etc.* as well as for computer
programs, the `index formulae' of Section 1.6 are more appropriate.

For orthonormal bases, because of * G* =

The formula for the angle between the
vectors (
) = **r** and
(
) = **t**

The volume of the unit cell

The volume of the unit cell of a crystal structure,
*i.e.* the body
containing all points with coordinates
, can be
calculated by the formula

In the general case one obtains

The formula (2.6.6) becomes simpler depending on the crystallographic
symmetry, *i.e.* on the crystal system.

**Copyright © 2002 International Union of
Crystallography**