Wiktionary
n. (context mathematics English) The equivalent of a determinant in matrix that have noncommutative entries
Wikipedia
In mathematics, the quasideterminant is a replacement for the determinant for matrices with noncommutative entries. Example 2 × 2 quasideterminants are as follows:
\left|\begin{array}{cc}
a_{11} & a_{12} \\
a_{21} & a_{22} \end{array}
\right|_{11} = a_{11} - a_{12}{a_{22}}^{-1}a_{21}
\qquad
\left|\begin{array}{cc}
a_{11} & a_{12} \\
a_{21} & a_{22} \end{array}
\right|_{12} = a_{12} - a_{11}{a_{21}}^{-1}a_{22}.
In general, there are n quasideterminants defined for an n × n matrix (one for each position in the matrix), but the presence of the inverted terms above should give the reader pause: they are not always defined, and even when they are defined, they do not reduce to determinants when the entries commute. Rather,
\left|A\right|_{ij} = (-1)^{i+j} \frac{\det A}{\det A^{ij}} ,
where A means delete the ith row and jth column from A.
The 2 × 2 examples above were introduced between 1926 and 1928 by Richardson and Heyting, but they were marginalized at the time because they were not polynomials in the entries of A. These examples were rediscovered and given new life in 1991 by I.M. Gelfand and V.S. Retakh. There, they develop quasideterminantal versions of many familiar determinantal properties. For example, if B is built from A by rescaling its i-th row (on the left) by ρ, then ∣B∣ = ρ∣A∣. Similarly, if B is built from A by adding a (left) multiple of the k-th row to another row, then ∣B∣ = ∣A∣ (∀j; ∀k ≠ i). They even develop a quasideterminantal version of Cramer's rule.