Linear Algebra/Definition of Determinant

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

$\left(\begin{array}{c}a\end{array}\right)$ is nonsingular iff $a\ne 0$

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

$\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ is nonsingular iff $ad-bc\ne 0$

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

$\left(\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right)$ is nonsingular iff $aei+bfg+cdh-hfa-idb-gec\ne 0$

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 $\underset{n\times n}{det}:{ℳ}_{n\times n}\to ℝ$ such that an $n\times n$ matrix $T$ is nonsingular if and only if $\underset{n\times n}{det}(T)\ne 0$. (We usually omit the subscript because if $T$ is $n\times n$ then "$det(T)$" could only mean "$\underset{n\times n}{det}(T)$".)