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Linear Algebra, Optimisation and Numerical Methods / Linear Transformations, Matrix Representation, and Rank-Nullity

Invertibility: Conditions and Computing A⁻¹ by Row Reduction

Topic 2 defined A−1A^{-1} mechanically, via Gauss-Jordan elimination on [A∣I][A\mid I]. This note closes Topic 4 by assembling every equivalent characterisation of invertibility developed across the unit so far — trivial kernel, full rank, nn pivots, nonzero determinant (previewed) — into one chain of "if and only if" statements, and by restating the computation in the language of linear maps rather than bare matrices: A−1A^{-1} is the matrix of the inverse transformation T−1T^{-1}, which exists exactly when TT is a bijection.
Learning objectives. By the end of this note you should be able to:
  • state the full list of equivalent conditions for invertibility and cite which topic/note proves each equivalence;
  • verify a matrix's invertibility using whichever condition is most convenient in context;
  • construct the inverse linear map T−1T^{-1} and confirm T−1∘T=idT^{-1}\circ T = \text{id};
  • explain why invertibility is a property of the transformation, not of any particular matrix representing it.

In this lesson

  1. The invertibility chain
  2. The inverse as a linear map

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