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3.2 Addition and Transposition 81
3.2 ADDITION AND TRANSPOSITION

In the previous chapters, matrix language and notation were used simply to for-
mulate some of the elementary concepts surrounding linear systems. The purpose
16
now is to turn this language into a mathematical theory.
Unless otherwise stated, a scalar is a complex number. Real numbers are
a subset of the complex numbers, and hence real numbers are also scalar quan-
tities. In the early stages, there is little harm in thinking only in terms of real
scalars. Later on, however, the necessity for dealing with complex numbers will
be unavoidable. Throughout the text, will denote the set of real numbers,
and C will denote the complex numbers. The set of all n -tuples of real numbers
n
will be denoted by , and the set of all complex n -tuples will be denoted
n 2
by C . For example, is the set of all ordered pairs of real numbers (i.e.,
3 m×n
the standard cartesian plane), and is ordinary 3-space. Analogously,
m×n
and C denote the m × n matrices containing real numbers and complex
numbers, respectively.
Matrices A =[a ij ] and B =[b ij ] are defined to be equal matrices
when A and B have the same shape and corresponding entries are equal. That
is, a ij = b ij for each i =1, 2,...,m and j =1, 2,...,n. In particular, this
 
1
definition applies to arrays such as u =   and v =( 1 2 3 ) . Even
2
3
though u and v describe exactly the same point in 3-space, we cannot consider
them to be equal matrices because they have different shapes. An array (or
matrix) consisting of a single column, such as u, is called a column vector,
while an array consisting of a single row, such as v, is called a row vector.

Addition of Matrices

If A and B are m × n matrices, the sum of A and B is defined to
be the m × n matrix A+B obtained by adding corresponding entries.
That is,

for each i and j.
[A + B] ij =[A] ij +[B] ij

For example,

−2 x 3 21 − x −2 0 1 1
+ = .
z +3 4 −y −34 + x 4+ y z 8+ x 4
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The great French mathematician Pierre-Simon Laplace (1749–1827) said that,“Such is the ad-
vantage of a well-constructed language that its simplified notation often becomes the source of
profound theories.” The theory of matrices is a testament to the validity of Laplace’s statement.