Matrices are frequently used when investigating the solutions of systems of linear equations. Decisive for the solubility and the possible number of solutions of such a system is a number, called the determinant or of A, which can be calculated for any square matrix A. In this section determinants are introduced and some of their laws are stated. Determinants are used to invert matrices and to calculate the volume of a unit cell in Subsections 2.6.1 and 2.6.3.
The theory of determinants is well developed and can be treated in a very general way. We only need determinants of (2 2) and (3 3) matrices and will discuss only these.
Definition (D 2.5.1)
Let
and
be a
(2 2) and a (3 3) matrix. Then their determinants are designated by
and are defined by the equations
Let D be a square matrix. If then the matrix D is called regular, if , then D is called singular. Here only regular matrices are considered. The matrix W of an isometry W is regular because always . In particular, holds.
Remark. The determinant is equal to the fraction , where is the volume of an original object and the volume of this object mapped by the affine mapping A. Isometries do not change distances, therefore they do not change volumes and holds.
The following rules hold for determinants of matrices A. The columns of A will be designated for these rules by .
Among these rules there are three procedures which do not change the value of the determinant:
is the product of the determinants.
The determinant of the product AB is
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