The volume of a simplex is given by the Cayley-Menger variety of this object, and the Chio condensation is a special case of one formula used to compute them, which states that this operation commutes when an identity element is added.  Hadamard put an upper bound on its value equal to n to the power of n over 2, and Sylvester's theorem about them is used to create Bayesian estimators for multivariable normal distributions.  When changing coordinate systems inside an integral this operation is applied to the Jacobian, and when finding the extrema of a function, this function is applied to the Hessian.  Used to show linear independence or orthogonality, this object is connected with the solution to a system of linear equations via Cramer's Rule. For 10 points, name this alternating multilinear operation applied to 
The volume of a simplex is given by the Cayley-Menger variety of this object, and the Chio condensation is a special case of one formula used to compute them, which states that this operation commutes when an identity element is added.  Hadamard put an upper bound on its value equal to n to the power of n over 2, and Sylvester's theorem about them is used to create Bayesian estimators for multivariable normal distributions.  When changing coordinate systems inside an integral this operation is applied to the Jacobian, and when finding the extrema of a function, this function is applied to the Hessian.  Used to show linear independence or orthogonality, this object is connected with the solution to a system of linear equations via Cramer's Rule. For 10 points, name this alternating multilinear operation applied to a matrix, which returns zero if the matrix is singular.
The volume of a simplex is given by the Cayley-Menger variety of this object, and the Chio condensation is a special case of one formula used to compute them, which states that this operation commutes when an identity element is added.  
{Det}erminant