The symplectic group is defined with these objects as its elements, as is the general linear group.
The Berezinian is an operation acting on the "super" form of these things. In representation theory, these objects are used in correspondence with group elements. The symplectic group is defined with these objects as its elements, as is the general linear group. In multivariable calculus, one of these objects is used in an analogue to the second derivative test, while another is used to evaluate integrals after a change of basis; those examples are the Hessian and Jacobian. Elementary operations can be used to put these objects in reduced row echelon form. Formulas derived from cofactor expansion, such as a times d minus b times c, compute these objects' determinants. For 10 points, identify these two-dimensional arrays of numbers.
The Berezinian is an operation acting on the "super" form of these things.
matrices [or matrix] &lt;Thompson&gt;