All matrices used are nonsingular and thus invertible. The elementary matrices generate the general linear group of invertible matrices. Groups are monoids for which every morphism is invertible. A groupoid is a small category in which every morphism is an isomorphism, i.e. invertible. An invertible endomorphism of is called an automorphism. An affine transformation is invertible if and only if formula 8 is invertible. The number of invertible matrices can be computed via the Chinese Remainder Theorem. Algebraic K-theory is related to the group of invertible matrices with coefficients the given field. The set of invertible distributions forms an abelian group under the convolution. This is impossible if "A" is nonsingular (invertible). Also, since formula 31 is nonsingular, it is invertible and therefore formula 40. A singular matrix is a square matrix which is not invertible. This structure is preserved by homeomorphisms, invertible maps that are continuous in both directions. The polar decomposition of an invertible matrix is unique. A matrix that is not invertible has condition number equal to infinity.