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 n2 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  means delete the ith row and jth column from A.

The  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 . 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  is built from  by rescaling its -th row (on the left) by , then . 
Similarly, if  is built from  by adding a (left) multiple of the -th row to another row, then . They even develop a quasideterminantal 
version of Cramer's rule.

Definition

Let  be an  matrix over a (not necessarily commutative) 
ring  and fix . Let 
denote the ()-entry of , let  denote the -th row of  with column  deleted, and let  denote the -th column of  with row  deleted. The ()-quasideterminant of  is defined if the submatrix  is invertible over . In this case, 

Recall the formula (for commutative rings) relating  to the determinant, namely . 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  matrix  with 
 a  matrix. If the ()-entry of  lies within , 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 ,  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 
 be the upper-left  submatrix of an 
 matrix  and fix a coordinate () in 
. Let  be the  matrix, with  defined as the ()-quasideterminant of the  matrix formed by adjoining to  the first  columns of row , the first  rows of column , and the entry . 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  to expand along. Recall the determinantal formula 
. 
Well, it happens that quasideterminants satisfy

(expansion along column ), and

(expansion along row ).

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.

See also 
 MacMahon Master theorem

References

Matrix theory
Determinants