As we saw previously, a matrix is a rectangular array of elements of
;
we say that the matrix is of
**order **
if it has *m* rows and *n* columns. The set of all matrices of order
with entries
in
is denoted
.

The matrix
is denoted
,
or
(*a*_{ij}) if no confusion is possible.

If *m*=*n*, the matrix is a square matrix of order *n*.

__Equality:__

Let
*A*=(*a*_{ij}) and
*B*=(*b*_{ij}) be two matrices in
.

.

__Addition:__

Let
*A*=(*a*_{ij}),
*B*=(*b*_{ij}) and
*C*=(*c*_{ij}) be three matrices in
.

__The zero matrix:__ the zero matrix in
is the matrix (*a*_{ij})such that
.
__Multiplication of a matrix by a scalar:__
Let
*A*=(*a*_{ij}) and
*B*=(*b*_{ij}) be two matrices in
and take
.

.

Let , and .

.

This definition is motivated by the link between matrices and linear operators: the product of two matrices corresponds to the composition of two linear operators (v.s. 1.4).