Matrices can be multiplied with a number or can be added, subtracted, and multiplied with each other. These operations obey the following rules:

**Definition** (D 2.4.5) An matrix * A* is multiplied with a
(real) number by multiplying each element with :

.

**Definition** (D 2.4.5) Let and be the general elements of the
matrices * A* and

*i.e.* the element of * C* is equal to the sum or
difference of the elements and of

The definition of matrix multiplication looks more complicated at first sight but it corresponds exactly to what is written in full in the formulae (2.2.1) to (2.2.4) of Section 2.2. The multiplication of two matrices is defined only if the number of columns of the ft trix is the same as the number of rows of the ght trix. The numbers of rows of the ft trix and of columns of the ght trix are free.

We first define the product of a matrix * A* with a column

**Definition** (D 2.4.5) The multiplication of an () matrix * A*
with an () column

Written as a matrix equation this is

In an analogous way one defines the multiplication of a row matrix with a general matrix.

**Definition** (D 2.4.5) The multiplication of a row * a*
,
with an () matrix

.

Written as a matrix equation this is

The multiplication of two matrices (both neither row nor column) is the combination of the already defined multiplications of a matrix with a column (matrix on the left, column on the right side) or of a row with a matrix (row on the left, matrix on the right side). Remember: The number of columns of the left matrix must be the same as the number of rows of the right matrix.

**Definition** (D 2.4.5) The *matrix product*
* C* =

is defined by

Examples.

If
and
,

then
. On the

other hand, .

Obviously, **C*** D*,

(

In `indices notation' (where * A* is an
matrix,

*Remarks*.

- The matrix
has the same number of columns as**A**has rows; the number of rows is `inherited' from**B**to**A**, the number of columns from**C**to**B**.**C** - A comparison with equation (2.2.4) shows that exactly the same
construction occurs in the matrix product when describing consecutive
mappings by matrix-column pairs. Also the product of the matrix
with the column**B**will be recognized. It is for this reason that the matrix formalism has been introduced. Affine mappings (also isometries and crystallographic symmetry operations) in point space are described by**a***matrix-column pairs*, see Sections 2.2 and 4.1. - The `power notation' is used in the same way for the matrix product of
a square matrix with itself as for numbers:
**A**=**A**;**A****A****A**=**A**,**A***etc.* - Using the formulae of this section one confirms equations (2.2.5)
to (2.2.8).

**Copyright © 2002 International Union of
Crystallography**