The invertible matrix theorem - Inverse of Matrices

The invertible matrix theorem


The Invertible Matrix Theorem states the following:
Let AA be a square n×nn \times n matrix. Then the following statements are equivalent. That is, for a given AA, the statements are either all true or all false.
1. AA is an invertible matrix.
2. AA is row equivalent to the n×nn \times n identity matrix.
3. AA has nn pivot positions.
4. The equation Ax=0Ax=0 has only the trivial solution.
5. The columns of AA form a linearly independent set.
6. The equation Ax=bAx=b has at least one solution for each bb in Rn\Bbb{R}^n.
7. The columns of AA span Rn\Bbb{R}^n.
8. The linear transformation xx Ax Ax maps Rn\Bbb{R}^n onto Rn\Bbb{R}^n.
9. There is an n×nn \times n matrix CC such that CA=ICA=I.
10. There is an n×nn \times n matrix DD such that AD=IAD=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.
  • Intro Lesson
    Characterizations of Invertible Matrices Overview:
  • 3.
    Understanding the Theorem
    Assume that AA is a square n×nn \times n matrix. Determine if the following statements are true or false:
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The invertible matrix theorem

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