Knowlpedia

Definition

Inverse matrix

A square matrix whose product with a given matrix on either side is the identity.

For a square AA over a field, an inverse matrix is a matrix A−1A^{-1} satisfying

A−1A=I=AA−1.A^{-1}A=I=AA^{-1}.

It exists exactly when the represented is , equivalently when det⁡A≠0\det A\ne0. It is unique: if BA=I=ACB A=I=A C, then B=B(AC)=(BA)C=CB=B(AC)=(BA)C=C.

Products and parameters

If A,BA,B are invertible then (AB)−1=B−1A−1(AB)^{-1}=B^{-1}A^{-1}. For a differentiable family of invertible matrices,

ddtA(t)−1=−A(t)−1A′(t)A(t)−1.\frac{d}{dt}A(t)^{-1}=-A(t)^{-1}A'(t)A(t)^{-1}.

This follows by differentiating A−1A=IA^{-1}A=I. Entrywise smoothness of the inverse also follows from the adjugate formula on the open set where the determinant is nonzero; uniform inverse bounds require quantitative control away from singular matrices.

References