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 "$\det\nolimits_{n \! \times \! n}(T)$".)