Page 88 - Matrix Analysis & Applied Linear Algebra
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82 Chapter 3 Matrix Algebra
The symbol “+” is used two different ways—it denotes addition between
scalars in some places and addition between matrices at other places. Although
these are two distinct algebraic operations, no ambiguities will arise if the context
in which “+” appears is observed. Also note that the requirement that A and
B have the same shape prevents adding a row to a column, even though the two
may contain the same number of entries.
The matrix (−A), called the additive inverse of A, is defined to be
the matrix obtained by negating each entry of A. That is, if A =[a ij ], then
−A =[−a ij ]. This allows matrix subtraction to be defined in the natural way.
For two matrices of the same shape, the difference A − B is defined to be the
matrix A − B = A +(−B) so that
[A − B] ij =[A] ij − [B] ij for each i and j.
Since matrix addition is defined in terms of scalar addition, the familiar algebraic
properties of scalar addition are inherited by matrix addition as detailed below.
Properties of Matrix Addition
For m × n matrices A, B, and C, the following properties hold.
Closure property: A + B is again an m × n matrix.
Associative property: (A + B)+ C = A +(B + C).
Commutative property: A + B = B + A.
Additive identity: The m × n matrix 0 consisting of all ze-
ros has the property that A + 0 = A.
Additive inverse: The m × n matrix (−A) has the property
that A +(−A)= 0.
Another simple operation that is derived from scalar arithmetic is as follows.
Scalar Multiplication
The product of a scalar α times a matrix A, denoted by αA, is defined
to be the matrix obtained by multiplying each entry of A by α. That
is, [αA] ij = α[A] ij for each i and j.
For example,
123 246 12 24
1
2 012 = 024 and 34 = 68 .
2
142 284 01 02
The rules for combining addition and scalar multiplication are what you
might suspect they should be. Some of the important ones are listed below.
