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Linear Algebra, Optimisation and Numerical Methods / Solving Linear Systems and Matrix Algebra

Inverse Matrices and Gauss-Jordan Elimination

"Undo" is a natural operation to want from a matrix: given the outcome Ax=bA\mathbf{x}=\mathbf{b} of a linear map, can you recover x\mathbf{x} directly from b\mathbf{b}? This note defines the inverse matrix A−1A^{-1} that does exactly that, states precisely when it exists, and shows how Gaussian elimination — pushed one step further than triangular form, all the way to the identity — computes it directly. The note closes on a caveat that matters in practice: computing A−1A^{-1} explicitly is rarely the fast way to solve Ax=bA\mathbf{x}=\mathbf{b}, a point the LULU derivation note that follows makes precise.
Learning objectives. By the end of this note you should be able to:
  • state the definition of A−1A^{-1} and the condition for its existence;
  • compute a 2×22\times2 inverse from the standard formula and a general inverse via Gauss-Jordan elimination on [A∣I][A\mid I];
  • prove that (AB)−1=B−1A−1(AB)^{-1}=B^{-1}A^{-1};
  • explain why solving via A−1bA^{-1}\mathbf{b} is not the practically preferred method.

In this lesson

  1. Definition and existence
  2. Computing A−1A^{-1} by Gauss-Jordan elimination
  3. Inverses and products
  4. Why A−1A^{-1} is rarely computed in practice

Before you start

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