Invertible Matrix Theorem -- From Wolfram MathWorld

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The invertible matrix theorem is a theorem in linear algebra which gives a series of equivalent conditions for an n×n square matrix A to have an inverse. In particular, A is invertible if and only if any (and hence, all) of the following hold:

1. A is row-equivalent to the n×n identity matrix I_n.

2. A has n pivot positions.

3. The equation Ax=0 has only the trivial solution x=0.

4. The columns of A form a linearly independent set.

5. The linear transformation x|->Ax is one-to-one.

6. For each column vector b in R^n, the equation Ax=b has a unique solution.

7. The columns of A span R^n.

8. The linear transformation x|->Ax is a surjection.

9. There is an n×n matrix C such that CA=I_n.

10. There is an n×n matrix D such that AD=I_n.

11. The transpose matrix A^(T) is invertible.

12. The columns of A form a basis for R^n.

13. The column space of A is equal to R^n.

14. The dimension of the column space of A is n.

15. The rank of A is n.

16. The null space of A is {0}.

17. The dimension of the null space of A is 0.

18. 0 fails to be an eigenvalue of A.

19. The determinant of A is not zero.

20. The orthogonal complement of the column space of A is {0}.

21. The orthogonal complement of the null space of A is R^n.

22. The row space of A is R^n.

23. The matrix A has n non-zero singular values.

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