Multiplying a matrix by one of these gives the identity matrix.
The existence of one of these entities corresponding to a function from R n to R n is guaranteed in some neighborhood if the Jacobian is nonsingular, according to a theorem often contrasted with the implicit function theorem. A generalization of one of them named for Moore and Penrose is an example of a "pseudo" one. The existence of these entities is only guaranteed for one of the operations in a ring, but for both operations in a field. If a function is injective, it has a corresponding one of these functions. A square matrix has one of these if and only if has a nonzero (*) determinant, and matrices without one of these objects are called singular. Multiplying a matrix by one of these gives the identity matrix. For 10 points, give this term whose multiplicative kind is the reciprocal and whose additive kind for a number x is negative x.
The existence of one of these entities corresponding to a function from R n to R n is guaranteed in some neighborhood if the Jacobian is nonsingular, according to a theorem often contrasted with the implicit function theorem.
inverse [accept inverse function or pseudoinverse or inverse element or inverse matrix]