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EQUIVALENT STATEMENTS If A is an n x n matrix, and TA : R n → R n is multiplication by A, then the following are equivalent. 1. A is invertible.  
Ax = 0 has only the trivial solution. 2. 3. The reduced row-­‐echelon form of A is I n . 4. A is expressible as a product of elementary matrices. 
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Ax = b is consistent for every n x 1 matrix b. 5. 
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Ax = b has exactly one solution for every n x 1 matrix b. 6. det ( A ) ≠ 0. 7. 8. The range of TA is R n . 9. TA 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 R n . 13. The row vectors of A span R n . 14. The column vectors of A form a basis for R n . 15. The row vectors of A form a basis for R n . 16. A has rank n. 17. 18. 19. A has nullity 0. The orthogonal complement of the nullspace of A is R n . 
The orthogonal complement of the row space of A is 0 . {}
20. AT A is invertible. 21. λ = 0 is not an eigenvalue of A.  
If Ax = b is a linear system of m equations in n unknowns, then the following are equivalent. 
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Ax = b is consistent for every m x 1 matrix b. 1. 2. The column vectors of A span R m . rank ( A ) = m. 3. If A is an m x n matrix, then the following are equivalent.  
Ax = 0 has only the trivial solution. 1. 2. The column vectors of A are LI. 
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Ax = b has at most one solution for every m x 1 matrix b. 3. AT A is invertible. 4. ************************************************************************