In mathematics, the quasideterminant is a replacement for the determinant for matrices with noncommutative entries. Example 2 × 2 quasideterminants are as follows:
| a 11 a 12 a 21 a 22 | 11 = a 11 − a 12 a 22 − 1 a 21 | a 11 a 12 a 21 a 22 | 12 = a 12 − a 11 a 21 − 1 a 22 . {\displaystyle \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 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,
| A | i j = ( − 1 ) i + j det A det A i j , {\displaystyle \left|A\right|_{ij}=(-1)^{i+j}{\frac {\det A}{\det A^{ij}}},}
where A i j {\displaystyle A^{ij}} means delete the ith row and jth column from A. The 2 × 2 {\displaystyle 2\times 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 {\displaystyle A} . 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 B {\displaystyle B} is built from A {\displaystyle A} by rescaling its i {\displaystyle i} -th row (on the left) by ρ {\displaystyle \left.\rho \right.} , then | B | i j = ρ | A | i j {\displaystyle \left|B\right|_{ij}=\rho \left|A\right|_{ij}} . Similarly, if B {\displaystyle B} is built from A {\displaystyle A} by adding a (left) multiple of the k {\displaystyle k} -th row to another row, then | B | i j = | A | i j ( ∀ j ; ∀ k ≠ i ) {\displaystyle \left|B\right|_{ij}=\left|A\right|_{ij}\,\,(\forall j;\forall k\neq i)} . They even develop a quasideterminantal version of Cramer's rule.
Definition
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