Page 430 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

If A is a matrix, then for every matrix b, the linear system is underdetermined. Thus must be

consistent for some b, and for each such b the general solution must have parameters, where r is the rank of A.

Summary

In Theorem 4.3.4 we listed eight results that are equivalent to the invertibility of a matrix A. We conclude this section by
adding eight more results to that list to produce the following theorem, which relates all of the major topics we have studied
thus far.

THEOREM 5.6.9

Equivalent Statements is multiplication by A, then the following are equivalent.
If A is an matrix, and if

(a) A is invertible.

(b) has only the trivial solution.

(c) The reduced row-echelon form of A is .

(d) A is expressible as a product of elementary matrices.
(e) is consistent for every matrix b.
(f) has exactly one solution for every matrix b.
(g) .

(h) The range of is .

(i) is one-to-one.

(j) The column vectors of A are linearly independent.
(k) The row vectors of A are linearly independent.
(l) The column vectors of A span .