Anon

Fundamentals of Matrix Algebra 395

Multiplication of a matrix by a scalar is associative with respect to
matrix addition:

cc()AB+=AB+c

Let’s now define the product of two matrices. Consider two matrices:

A={}ait np and B={}bsj pm

The product C = AB is defined as follows:

=={}= ∑















= 

CAB caij itbtj

t

p

1

The product C = AB is therefore a matrix whose generic element {cij}
is the scalar product of the ith row of the matrix A and the jth column
of the matrix B. This definition generalizes the definition of scalar prod-
uct of vectors: the scalar product of two n-dimensional vectors is the
product of an n × 1 matrix (a row vector) for a 1 × n matrix (the column
vector).

inverse and Adjoint Consider two square matrices of order n A and B. If AB =
BA = I, then the matrix B is called the inverse of A and is denoted as A–1. It
can be demonstrated that the two following properties hold:

Property 1. A square matrix A admits an inverse A–1 if and only if

it is nonsingular, that is, if and only if its determinant is different

from zero. Otherwise stated, a matrix A admits an inverse if and

only if it is of full rank.

Property 2. The inverse of a square matrix, if it exists, is unique.

This property is a consequence of the property that, if A is nonsin-

gular, then AB = AC implies B = C.

Consider now a square matrix of order n, A = {aij} and consider its
cofactors αij. Recall that the cofactors αij are the signed minors

()−

1 ()ij+ M

ij