Page 161 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

If , then 9 becomes

Solving this system yields (verify) , , so the eigenvectors corresponding to are the nonzero solutions
of the form

Again from 9, the eigenvectors of A corresponding to are the nontrivial solutions of

We leave it for the reader to solve this system and show that the eigenvectors of A corresponding to are the nonzero
solutions of the form

Summary

In Theorem 1.6.4 we listed five results that are equivalent to the invertibility of a matrix . We conclude this section by
merging Theorem 2.3.3 with that list to produce the following theorem that relates all of the major topics we have studied
thus far.
THEOREM 2.3.6

Equivalent Statements
If A is an matrix, then the following statements are equivalent.

(a) A is invertible.

(b) has only the trivial solution.

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

(d) A can be expressed as a product of elementary matrices.

(e) is consistent for every matrix .

(f) has exactly one solution for every matrix .

(g) .