Linear Algebra, 2020a

326 Chapter Four. Determinants
I
Definition
Determining nonsingularity is trivial for 1×1 matrices.
( )
a is nonsingular iff a ≠ 0
Corollary Three.IV.4.11 gives the 2×2 formula.
( )
a b
is nonsingular iff ad − bc ≠ 0
c d
We can produce the 3×3 formula as we did the prior one, although the computation
is intricate (see Exercise 10).
⎛
⎜
a b c
⎞
⎟
⎝d e f⎠ is nonsingular iff aei + bfg + cdh − hfa − idb − gec ≠ 0
g h i
With these cases in mind, we posit a family of formulas: a, ad−bc, etc. For each
n the formula defines 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 n×n because the size of T describes which determinant function
we mean.)
I.1 Exploration
This subsection is an optional motivation and development of the general
definition. The definition is in the next subsection.
Above, in each case the matrix is nonsingular if and only if some formula is
nonzero. But the three formulas don’t show an obvious pattern. We may spot
that the 1×1 term a has one letter, that the 2×2 terms ad and bc have two
letters, and that the 3×3 terms each have three letters. We may even spot that
in those terms there is a letter from each row and column of the matrix, e.g., in
the cdh term one letter comes from each row and from each column.
⎛ ⎞
⎜
⎝d
h
But these observations are perhaps more puzzling than enlightening. For instance,
we might wonder why some terms are added but some are subtracted.
c
⎟
⎠