In the noncommutative case, determinants are not useful invariants of matrices (in fact, various classical formulas for determinants mutually disagree over noncommutative rings) and other polynomial suggestions were not of much success (in some cases the superdeterminant and Dieudonné’s determinant are of use, but they can be easily expressed in terms of quasideterminants anyway). Quasideterminants will be noncommutative rational functions, rather than polynomial, expressions.
Let $A = (a^i_j)\in M_n(R)$ be an $n\times n$ matrix over an arbitrary noncommutative (but unital and associative) ring $R$. In fact it makes sense to work with many objects (see horizontal categorification): having, say, an abelian category where $a^i_j$ is a morphism from the object $i$ to the object $j$. Let us choose a row label $i$ and a column label $j$. By $A^{\hat{i}}_{\hat{j}}$ we’ll denote the $(n-1)\times(n-1)$ matrix obtained from $A$ by removing the $i$-th row and the $j$-th column. The $(i,j)$-th quasideterminant $|A|_{ij}$ is
provided the right-hand side is defined (the corresponding inverses exist).
Up to $n^2$ quasideterminants of a given $A \in M_n(R)$ may be defined. If all the $n^2$ quasideterminants $|A|_{ij}$ exist and are invertible then the inverse $A^{-1}$ of $A$ exists in $M_n(R)$ and
Quasideterminants for a matrix with entries in a commutative ring $R$ are, up to a sign, ratios of the determinant of the matrix and the determinant of the appropriate $(n-1)\times (n-1)$ submatrix, as one can see by remembering the formula for matrix inverse in terms of the cofactor matrices. For systems of linear equations with coefficients (from one side) in a noncommutative ring, there are solution formulas involving quasideterminants and generalizing Cramer’s rule (hence left and right Cramer’s rule). Quasideterminants with generic entries (entries in a free skewfield) satisfy generalizations of many classical identities: Muir’s law of extensionality, Silvester’s law, etc., and a new “heredity” principle and so-called “homological identities”. Furthermore, for polynomial equations over noncommutative rings, noncommutative Viete’s formulas have been found and used in applications.
Quasideterminants were introduced by I. Gel’fand and V. Retakh around 1990.
I. M. Gel'fand, V. S. Retakh, Determinants of matrices over noncommutative rings, Funct.Anal.Appl. 25 (1991), no.2, pp. 91–102. engl. transl. 21 (1991), pp. 51–58.
I.M. Gel’fand, V.S. Retakh, A theory of noncommutative determinants and characteristic functions of graphs, Funct.Anal.Appl. 26 (1992), no.4, pp. 231–246.
I.M. Gel’fand, V.S. Retakh, Quasideterminants I, Selecta Mathematica, New Series 3 (1997) no.4, pp. 517–546;
D.Krob, B.Leclerc, Minor identities for quasi-determinants and quantum determinants, Comm.Math.Phys. 169 (1995), pp. 1–23.
Z. Škoda, Noncommutative localization in noncommutative geometry (Chapter 16: Quasideterminants and Cohn localization), London Math. Society Lecture Note Series 330, ed. A. Ranicki; pp. 220–313, arXiv:math.QA/0403276)
V. Retakh, R. Wilson, Advanced course on quasideterminants and universal localization, pdf