Page 315 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition
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(9)
If , this system is
which has the solutions , , (verify), or, in matrix form,
As anticipated, these are the vectors along the z-axis. If , then system 9 is
which has the solutions , , (verify), or, in matrix form,
As anticipated, these are the vectors in the -plane.
Summary
In Theorem 2.3.6 we listed six results that are equivalent to the invertibility of a matrix A. We conclude this section by merging
Theorem 4.3.1 with that list to produce the following theorem that relates all of the major topics we have studied thus far.
THEOREM 4.3.4
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.
