For a matrix A and column vectors x and b, A must have this property to solve "A x equals b.
Localizing a subset S of a ring gives every element of S this property. For an integral domain R, taking "R cross R" and quotienting out by "a b" is equivalent to "x y" if "a times y equals b times x" gives almost every element this property, producing Frac R. The contraction mapping theorem can be used to prove a local form of this property around a point where the Jacobian is nonzero. The group axioms specify closure, associativity, the existence of an identity, and that every element has this property. For a matrix A and column vectors x and b, A must have this property to solve "A x equals b." A function has this property if and only if it is bijective. A matrix with a non-zero determinant has this property. For 10 points, if a function f of x equals y, what property requires the existence of a function g such that g of y equals x?
Localizing a subset S of a ring gives every element of S this property.
invertiblity [or invertible; or non-singular; accept equivalents such as having an inverse]