EQUIVALENT STATEMENTS

If *A* is an *n x n *matrix, and if
is multiplication by *A*, then the
following are equivalent.

1. *A* is invertible.

2. has only the trivial solution.

3. The
reduced row-echelon form of *A* is

4. *A* is expressible as a product of elementary matrices.

5. is consistent for
every *n x 1 *matrix

6. has exactly one
solution for every *n x 1 *matrix

7.

8. The range of

9. is *1 – 1*.

10. The
column vectors of *A* are LI.

11. The
row vectors of *A* are LI.

12. The
column vectors of *A* span

13. The
row vectors of *A* span

14. The
column vectors of *A* form a basis
for

15. The
row vectors of *A* form a basis for

16. *A* has rank *n*.

17. *A* has nullity 0.

18. The
orthogonal complement of the nullspace of *A* is

19. The
orthogonal complement of the row space of *A* is

20. is invertible.

21. is not an eigenvalue
of *A*.

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If is a linear system of *m* equations in *n* unknowns, then the following are equivalent.

1. is consistent for
every *m x 1* matrix

2. The
column vectors of *A* span

3.

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If *A* is an *m x n *matrix, then the following are equivalent.

1. has only the trivial solution.

2. The
column vectors of *A* are LI.

3. has at most one
solution for every *m x 1* matrix

4. is invertible.

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