194 CHAPTER 9. LINEAR TRANSFORMATIONS

what is given to be so, it follows (AB)A = A and so using the associative law for matrixmultiplication,

A(BA)−A = A(BA− I) = 0.

But this means (BA− I)x = 0 for all x since otherwise, A would not be one to one. HenceBA = I as claimed. ■

This theorem shows that if an n× n matrix B acts like an inverse when multiplied onone side of A it follows that B = A−1and it will act like an inverse on both sides of A.

The conclusion of this theorem pertains to square matrices only. For example, let

A =

 1 00 11 0

 , B =

(1 0 01 1 −1

)(9.2)

Then

BA =

(1 00 1

)but

AB =

 1 0 01 1 −11 0 0

 .

There is also an important characterization in terms of determinants. This is provedcompletely in the section on the mathematical theory of the determinant.

Theorem 9.2.9 Let A be an n× n matrix and let TA denote the linear transformation de-termined by A. Then the following are equivalent.

1. TA is one to one.

2. TA is onto.

3. det(A) ̸= 0.

9.2.5 The General Solution Of A Linear SystemRecall the following definition which was discussed above.

Definition 9.2.10 T is a linear transformation if for x,y vectors and a,b scalars,

T (ax+by) = aT x+bT y. (9.3)

Thus linear transformations distribute across addition and pass scalars to the outside. Alinear system is one which is of the form

T x = b.

If T xp = b, then xp is called a particular solution to the linear system.