MULTIPLICATION OF MATRICES
The multiplication of two matrices and is possible only if the number
of columns of is equal to the number of rows of ; that is, if is an
(mk) matrix and is a (kn) matrix. Such matrices are said to be con-
formable. Then the product
(E.9)
is an (mn) matrix, where
PROPERTIES OF MATRIX PRODUCTS
It is mentioned that, in general, the product However, for
conformable matrices, multiplication is associative; that is,
(E.10)
And it is distributive; that is,
(E.11)
From the definition of the transpose of a matrix, it follows that
(E.12)
Also, the following product plays an important role in Chapter 7. Let {x} be an
(n1) column vector and be a square matrix of order n. Then the product
(E.13)
is a scalar, since the product results in a (1n) matrix and, therefore,
the product results in a (11) matrix, or a scalar.
MATRIX INVERSE
The inverse of a square matrix , which is denoted as is defined as
that matrix for which
(E.14)
The inverse of a matrix exists if
where denotes the determinant of .3A 4det3A4
det 3A 4 0
3A 4
1
3A 4 3A43A4
1
3I 4
3A 4
1
,3A 4
5x6
T
3A 45x6
5x6
T
3A 4
c 5x6
T
3A 45x6
3A 4
13A43B 42
T
3B 4
T
3A 4
T
13A4 3B 423C 4 3A43C4 3B 43C 4
13C43A 423B 4 3C413A43B 42
3A 43B 4 3B43A4.
c
ij
a
k
p1
a
ip
b
pj
3C 4 3A43B4
3B 4
3A 43B43A 4
3B 43A4
Matrix Inverse 677