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