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Linear Algebra, Definition of Determinant

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For {}1 \ × \ 1 matrices, determining nonsingularity
is trivial.

( \begin{matrix}a\end{matrix} )

is nonsingular iff
a ≠ 0

The {}2 \ × \ 2 formula came out in the course of developing
the inverse.

( \begin{matrix}a & b \\ c & d\end{matrix} )

is nonsingular iff
a d − b c ≠ 0

The {}3 \ × \ 3 formula can be produced similarly
(see Problem 9).

( \begin{matrix}a & b & c \\ d & e & f \\ g & h & i\end{matrix} )

is nonsingular iff
a e i + b f g + c d h − h f a − i d b − g e c ≠ 0

With these cases in mind, we posit a family of
formulas, a, a d − b c, etc.
For each n the formula gives rise to a
determinant function
{det}_{n \ × \ n} : {ℳ}_{n \ × \ n} → ℝ
such that an n \ × \ n matrix T is nonsingular if and
only if {det}_{n \ × \ n} ( T ) ≠ 0.
(We usually omit the subscript because
if T is n \ × \ n then "det ⁡ ( T )"
could only mean "{det}_{n \ × \ n} ( T )".)

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