GraphMath

Inverse

Definition, computation, uniqueness, properties and applications

When does a matrix have a true inverse and what does that inverse do?

This chapter presents the inverse matrix through algebraic and geometric definition, row-reduction computation, uniqueness, existence conditions, algebraic properties, inverse of a product and solving A×x = b.

Open chapter PDF Key ideas Chapter index Back to Linear Algebra

Key ideas

An inverse exists only when a square matrix preserves enough information to be fully reversed.

  • M⁻¹ is defined by M⁻¹M = MM⁻¹ = I, so it reverses the transformation by M
  • Row reduction of [ M | I ] computes the inverse by turning M into I and carrying along the transformation on the identity block
  • A true inverse is unique, so a full-rank square matrix cannot have two different inverses
  • Inverse exists only for square full-rank matrices: if rank is deficient, some information is lost and cannot be recovered
  • The inverse of a product reverses the order: (AB)⁻¹ = B⁻¹A⁻¹

The chapter ends by interpreting inverse as the solution map for A×x = b and by giving the trivial 1×1 and 2×2 formulas.

Why must inverse reverse the order in a product?

Because AB means apply B first and then A. To undo that combined transformation, we must first undo A and then undo B. That is why (AB)⁻¹ equals B⁻¹A⁻¹ rather than A⁻¹B⁻¹.

Related chapters

  • Row reduction explains the elimination process and elementary matrices used here to compute M⁻¹ from [ M | I ]
  • Linear transformations gives the geometric language behind “undoing” a map and interpreting inverse as reversal of a transformation
  • Rank supports the full-rank requirement and the failure of inverse in rank-deficient cases
  • Linear equations connects inverse to solving A×x = b and uniqueness of solutions
  • Inverse & Pseudoinverse extends the true inverse to tall, wide and rank-deficient matrices where a full inverse does not exist

Chapter contents

The PDF is a single document. The page links below are best-effort: most browsers support them, but some viewers may ignore the page hint.

Topic Pages
Algebraic & geometric definition 1
Computing M⁻¹ by row reduction of [ M | I ] 2–3
Uniqueness of M⁻¹ 3–4
M⁻¹ is defined only for square full-rank M 4–5
Algebraic properties of M⁻¹ 6–7
Visual example of (AB)⁻¹ 7–8
M⁻¹ as solution to A×x = b 8–10
Trivial cases: 1×1 and 2×2 10
Open PDF

Why does a rank-deficient square matrix fail to have an inverse?

Because rank deficiency means some input directions are collapsed or some output directions are unreachable. Once information is lost, no matrix can recover it for every possible target.

Was this chapter helpful?

Quick feedback helps us improve the site.

Yes No