6 Aug 2016


Submitted by Rob Ireton
using the fold-out screen to get this without lying prone on the floor
21 Jul 2016

Invertible Matrix Theorem

Submitted by Rob Ireton
If 𝐴 is an 𝑛 × 𝑛 matrix, then the following statements are logically equivalent:
  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 are linearly independent.
  6. The linear transformation 𝐱 ⟼ 𝐴𝐱 is one-to-one.
  7. The equation 𝐴𝐱 = 𝐛 has at least one solution for each 𝐛 in ℝ𝑛.


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