For matrices, determining nonsingularity is trivial.

is nonsingular iff

The formula came out in the course of developing the inverse.

is nonsingular iff

The formula can be produced similarly (see Problem 9).

is nonsingular iff

With these cases in mind, we posit a family of formulas, , , etc. For each the formula gives rise to a **determinant** function such that an matrix is nonsingular if and only if . (We usually omit the subscript because if is then "" could only mean "".)