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 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 ${\displaystyle A^{ij}}$ means delete the ith row and jth column from A.

The ${\displaystyle 2\times 2}$ 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 ${\displaystyle A}$. 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 ${\displaystyle B}$ is built from ${\displaystyle A}$ by rescaling its ${\displaystyle i}$-th row (on the left) by ${\displaystyle \left.\rho \right.}$, then ${\displaystyle \left|B\right|_{ij}=\rho \left|A\right|_{ij}}$. Similarly, if ${\displaystyle B}$ is built from ${\displaystyle A}$ by adding a (left) multiple of the ${\displaystyle k}$-th row to another row, then ${\displaystyle \left|B\right|_{ij}=\left|A\right|_{ij}\,\,(\forall j;\forall k\neq i)}$. They even develop a quasideterminantal version of Cramer's rule.

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