Page 122 - Matrix Analysis & Applied Linear Algebra

116 Chapter 3 Matrix Algebra

Although not all matrices are invertible, when an inverse exists, it is unique.
To see this, suppose that X 1 and X 2 are both inverses for a nonsingular matrix
A. Then
X 1 = X 1 I = X 1 (AX 2 )=(X 1 A)X 2 = IX 2 = X 2 ,
which implies that only one inverse is possible.
Since matrix inversion was defined analogously to scalar inversion, and since
matrix multiplication is associative, exactly the same reasoning used in (3.7.1)
and (3.7.2) can be applied to a matrix equation AX = B, so we have the
following statements.

Matrix Equations

• If A is a nonsingular matrix, then there is a unique solution for X
in the matrix equation A n×n X n×p = B n×p , and the solution is

X = A −1 B. (3.7.4)

• A system of n linear equations in n unknowns can be written as a
single matrix equation A n×n x n×1 = b n×1 (see p. 99), so it follows
from (3.7.4) that when A is nonsingular, the system has a unique
solution given by x = A −1 b.

However, it must be stressed that the representation of the solution as
x = A −1 b is mostly a notational or theoretical convenience. In practice, a
nonsingular system Ax = b is almost never solved by first computing A −1 and
then the product x = A −1 b. The reason will be apparent when we learn how
much work is involved in computing A −1 .
Since not all square matrices are invertible, methods are needed to distin-
guish between nonsingular and singular matrices. There is a variety of ways to
describe the class of nonsingular matrices, but those listed below are among the
most important.

Existence of an Inverse
For an n × n matrix A, the following statements are equivalent.

• A −1 exists (A is nonsingular). (3.7.5)
• rank (A)= n. (3.7.6)
Gauss–Jordan
• A −−−−−−−−→ I. (3.7.7)
• Ax = 0 implies that x = 0. (3.7.8)