Linear Algebra/Definition of Determinant

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

${\begin{pmatrix}a\end{pmatrix}}$ is nonsingular iff $a\neq 0$

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

${\begin{pmatrix}a&b\\c&d\end{pmatrix}}$ is nonsingular iff $ad-bc\neq 0$

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

${\begin{pmatrix}a&b&c\\d&e&f\\g&h&i\end{pmatrix}}$ is nonsingular iff $aei+bfg+cdh-hfa-idb-gec\neq 0$

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