Linear Algebra/Definition of Determinant

For ${\displaystyle 1\!\times \!1}$ matrices, determining nonsingularity is trivial.

The ${\displaystyle 2\!\times \!2}$ formula came out in the course of developing the inverse.

The ${\displaystyle 3\!\times \!3}$ formula can be produced similarly (see Problem 9).

With these cases in mind, we posit a family of formulas, ${\displaystyle a}$, ${\displaystyle ad-bc}$, etc. For each ${\displaystyle n}$ the formula gives rise to a determinant function ${\displaystyle \det \nolimits _{n\!\times \!n}:{\mathcal {M}}_{n\!\times \!n}\to \mathbb {R} }$ such that an ${\displaystyle n\!\times \!n}$ matrix ${\displaystyle T}$ is nonsingular if and only if ${\displaystyle \det \nolimits _{n\!\times \!n}(T)\neq 0}$. (We usually omit the subscript because if ${\displaystyle T}$ is ${\displaystyle n\!\times \!n}$ then "${\displaystyle \det(T)}$" could only mean "${\displaystyle \det \nolimits _{n\!\times \!n}(T)}$".)