Download 4.19.1. Theorem 4.20

4.19.1. Theorem 4.20
Definition:
Nonsingular Matrix
Let A   aij  be a square n  n matrix.
If there exists another n  n matrix B such
that BA  I , where I is the n  n identity matrix, the A is said to be nonsingular and
B is called the left-inverse of A.
Theorem 4.20.
(a) If B is a left inverse of an n  n matrix A, then it is also a right inverse, i.e.,
AB  I . Furthermore, B is unique.
(b) If B is a right inverse of an n  n matrix A, then it is also a left inverse.
A is nonsingular.
Hence,
Proof of (a)
Consider the linear transformation T : R n  R n such that m T   A   aij 
relative to the basis of unit coordinate vectors.
Given x such that T  x   O , let X be the n  1 column matrix that corresponds to x.
We have AX  0 , where 0 is the zero column matrix. Thus, B  AX   0 for any n
 n matrix B. If B is a left inverse of A, then
B  AX    BA X  I X  X  0
This means T  x   O implies x  O . Hence, by theorems 4.8 and 4.10, T is
invertible so that there is a unique left inverse T 1 , which is also a right inverse.
Matrix representation of the equation TT 1  T 1T  I then gives
A m T 1   m T 1  A  I
Multiplying by the left inverse B, we have
m  T 1   B
so that
AB  I
i.e., B is also a right inverse. Finally, let C be another left inverse so that
CA  I
Multiplying this on the right by B gives C  B so that the left inverse is unique.
QED.
Proof of (b)
The condition AB  I that B is a right inverse of A can be interpreted as saying B
has a left inverse A. According to (a), A is also a right inverse of B so that BA  I .
QED.