One of these, a polynomial calculated from a similarly named object consisting of geometric progressions, is named after Vandermonde. The scaling factor for a change of coordinates when calculating double or triple integrals is the Jacobian example of this quantity. If this quantity is equal to 1 for a unitary operator, it is a rotation. For a given operator, this quantity is equal to the (*) product of its eigenvalues. A system of equations can be solved by the quotient of two of these by Cramer's rule, and the cross product is calculated using one of these. For 10 points, name this quantity 
One of these, a polynomial calculated from a similarly named object consisting of geometric progressions, is named after Vandermonde. The scaling factor for a change of coordinates when calculating double or triple integrals is the Jacobian example of this quantity. If this quantity is equal to 1 for a unitary operator, it is a rotation. For a given operator, this quantity is equal to the (*) product of its eigenvalues. A system of equations can be solved by the quotient of two of these by Cramer's rule, and the cross product is calculated using one of these. For 10 points, name this quantity of a square matrix indicated by straight lines around it, which for 2-by-2 matrices is calculated as ad minus bc.
One of these, a polynomial calculated from a similarly named object consisting of geometric progressions, is named after Vandermonde.
determinant