Quasideterminant

In mathematics, the quasideterminant is a replacement for the determinant for matrices with noncommutative entries. Example 2 × 2 quasideterminants are as follows:
In general, there are "n"2 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,
where formula_3 means delete the "i"th row and "j"th column from "A".
The formula_4 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 formula_5. These examples were rediscovered and given new life in 1991 by Israel Gelfand and Vladimir Retakh. There, they develop quasideterminantal versions of many familiar determinantal properties. For example, if formula_6 is built from formula_5 by rescaling its formula_8-th row (on the left) by formula_9, then formula_10. 
Similarly, if formula_6 is built from formula_5 by adding a (left) multiple of the formula_13-th row to another row, then formula_14. They even develop a quasideterminantal 
version of Cramer's rule.
Definition.
Let formula_5 be an formula_16 matrix over a (not necessarily commutative) 
ring formula_17 and fix formula_18. Let formula_19
denote the (formula_20)-entry of formula_5, let formula_22 denote the formula_8-th row of formula_5 with column formula_25 deleted, and let formula_26 denote the formula_25-th column of formula_5 with row formula_8 deleted. The (formula_20)-quasideterminant of formula_5 is defined if the submatrix formula_3 is invertible over formula_17. In this case, 
Recall the formula (for commutative rings) relating formula_35 to the determinant, namely formula_36. The above definition is a generalization in that (even for noncommutative rings) one has
whenever the two sides makes sense.
Identities.
One of the most important properties of the quasideterminant is what Gelfand and Retakh 
call the "heredity principle". It allows one to take a quasideterminant in 
stages (and has no commutative counterpart). To illustrate, suppose
is a block matrix decomposition of an formula_16 matrix formula_5 with 
formula_41 a formula_42 matrix. If the (formula_20)-entry of formula_5 lies within formula_41, it says that 
That is, the quasideterminant of a quasideterminant is a quasideterminant. To put it less succinctly: UNLIKE determinants, quasideterminants treat matrices with block-matrix entries no differently than ordinary matrices (something determinants cannot do since block-matrices generally don't commute with one another). That is, while the precise form of the above identity is quite surprising, the existence of "some" such identity is less so. 
Other identities from the papers are (i) the so-called "homological relations", stating that two quasideterminants in a common row or column are closely related to one another, and (ii) the Sylvester formula.
(i) Two quasideterminants sharing a common row or column satisfy
or
respectively, for all choices formula_49, formula_50 so that the 
quasideterminants involved are defined.
(ii) Like the heredity principle, the Sylvester identity is a way to recursively compute a quasideterminant. To ease notation, we display a special case. Let 
formula_51 be the upper-left formula_42 submatrix of an 
formula_53 matrix formula_5 and fix a coordinate (formula_20) in 
formula_51. Let formula_57 be the formula_58 matrix, with formula_59 defined as the (formula_60)-quasideterminant of the formula_61 matrix formed by adjoining to formula_51 the first formula_13 columns of row formula_64, the first formula_13 rows of column formula_66, and the entry formula_67formula_68. Then one has
Many more identities have appeared since the first articles of Gelfand and Retakh on the subject, most of them being analogs of classical determinantal identities. An important source is Krob and Leclerc's 1995 article. To highlight one, we consider the row/column expansion identities. Fix a row formula_8 to expand along. Recall the determinantal formula 
formula_71. 
Well, it happens that quasideterminants satisfy
(expansion along column formula_25), and
(expansion along row formula_8).
Connections to other determinants.
The quasideterminant is certainly not the only existing determinant analog for noncommutative settings—perhaps the most famous examples are the Dieudonné determinant and quantum determinant. However, these are related to the quasideterminant in some way. For example,
with the factors on the right-hand side commuting with each other. Other famous examples, such as Berezinians, Moore and Study determinants, Capelli determinants, and Cartier-Foata-type determinants are also expressible in terms of quasideterminants. Gelfand has been known to define a (noncommutative) determinant as "good" if it may be expressed as products of quasiminors.
Applications.
Paraphrasing their 2005 survey article with Sergei Gelfand and Robert Wilson 
Israel Gelfand and Vladimir Retakh advocate for the adoption of quasideterminants as "a main organizing tool in noncommutative algebra, giving them the same role determinants play in commutative algebra." Substantive use has been made of the quasideterminant in such fields of mathematics as integrable systems, representation theory, algebraic combinatorics, the theory of "noncommutative symmetric functions", the theory of "polynomials over division rings", and noncommutative geometry.
Several of the applications above make use of "quasi-Plücker coordinates," which parametrize noncommutative Grassmannians and flags in much the same way as Plücker coordinates do Grassmannians and flags over commutative fields. More information on these can be found in the survey article.