This alternating form is also a multiplicative map, and the Cayley-Hamilton theorem relates this to a polynomial.
Friedholm names a generalization of this to operators, while DieudonnÃ© names a generalization to non-commutative spaces. Another generalization of this is called the Berezinian and is used on supermanifolds. Jacobi's formula is used to di erentiate this, and Sylvester's theorem relates two instances of this to each other. If this quantity for some object is equal to 1 or −1, then that object is referred to as unimodular. This alternating form is also a multiplicative map, and the Cayley-Hamilton theorem relates this to a polynomial. Laplace's for- mula for calculating this quantity involves multiplying the input by its adjugate. For ten points, identify this quantity that is zero for non-invertible matrices. 
Friedholm names a generalization of this to operators, while DieudonnÃ© names a generalization to non-commutative spaces.
determinant