The Invertible Matrix Theorem states the following:
Let A be a square n×n matrix. Then the following statements are equivalent. That is, for a given A, the statements are either all true or all false.
1. A is an invertible matrix.
2. A is row equivalent to the n×n identity matrix.
3. A has n pivot positions.
4. The equation Ax=0 has only the trivial solution.
5. The columns of A form a linearly independent set.
6. The equation Ax=b has at least one solution for each b in Rn.
7. The columns of A span Rn.
8. The linear transformation x→Ax maps Rn onto Rn.
9. There is an n×n matrix C such that CA=I.
10. There is an n×n matrix D such that AD=I.
There are extensions of the invertible matrix theorem, but these are what we need to know for now. Keep in mind that this only works for square matrices.
Characterizations of Invertible Matrices Overview:
Understanding the Theorem
Assume that A is a square n×n matrix. Determine if the following statements are true or false:
The invertible matrix theorem
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