Theory of Determinant

4. Generalization

The determinant can be defined through multilinear and alternating properties. By extending the properties to nonsquare matrices, the concept of determinant can be extended.

For the 2 by 3 case

 A = [ a11 a12 a13 ] = [a11e1 + a21e2, a12e1 + a22e2, a13e1 + a23e2], a21 a22 a23

the multilinearity implies

D(A) = D(a11e1 + a21e2, a12e1 + a22e2, a13e1 + a23e2)
= a11D(e1, a12e1 + a22e2, a13e1 + a23e2) + a21D(e2, a12e1 + a22e2, a13e1 + a23e2)
= a11a12D(e1, e1, a13e1 + a23e2) + a11a22D(e1, e2, a13e1 + a23e2)
+ a21a12D(e2, e1, a13e1 + a23e2) + a21a22D(e2, e2, a13e1 + a23e2)
= a11a12a13D(e1, e1, e1) + a11a12a23D(e1, e1, e2)
+ a11a22a13D(e1, e2, e1) + a11a22a23D(e1, e2, e2)
+ a21a12a13D(e2, e1, e1) + a21a12a23D(e2, e1, e2)
+ a21a22a13D(e2, e2, e1) + a21a22a23D(e2, e2, e2).

Note that in each D(ei, ej, ek), at least two from i, j, k are equal. Therefore the alternating property would further imply D(ei, ej, ek) = 0. We conclude that any multilienar and alternating function of 2 by 3 matrices must be zero.

Similarly, for the 3 by 2 case

 A = [ a11 a12 ] = [a11e1 + a21e2 + a31e3, a12e1 + a22e2 + a32e3], a21 a22 a31 a32

the multilinearity implies

D(A) = D(a11e1 + a21e2 + a31e3, a12e1 + a22e2 + a32e3)
= a11D(e1, a12e1 + a22e2 + a32e3)
+ a21D(e2, a12e1 + a22e2 + a32e3)
+ a31D(e3, a12e1 + a22e2 + a32e3)
= a11a12D(e1, e1) + a11a22D(e1, e2) + a11a32D(e1, e3)
+ a21a12D(e2, e1) + a21a22D(e2, e2) + a21a32D(e2, e3)
+ a31a12D(e3, e1) + a31a22D(e3, e2) + a31a32D(e3, e3).

The alternating property further implies

D(A) = (a11a22 - a21a12)D(e1, e2) + (a11a32 - a31a12)D(e1, e3) + (a21a32 - a31a22)D(e2, e3)

 = D(e1, e2) det[ a11 a12 ] + D(e1, e3) det[ a11 a12 ] + D(e2, e3) det[ a21 a22 ]. a21 a22 a31 a32 a31 a32

Thus the function D(A) is determined by three numbers D(e1, e2), D(e1, e3), D(e2, e3). The three numbers can actually be arbitrarily chosen, because for any numbers D12, D13, D23, the function

 D(A) = D12det[ a11 a12 ] + D13det[ a11 a12 ] + D23det[ a21 a22 ] a21 a22 a31 a32 a31 a32

is multilinear and alternating. A fancy way of saying this is that all the multilinear and alternating functions on 3 by 2 matrices form a vector space of dimension 3, with the three 2 by 2 determinants as a basis.

In general, for m by n matrix A, the multilinearity gives us (see here)

D(A) = ∑1≤i1,i2,...,inm ai11ai22...ainn D(ei1, ei2, ..., ein).

By the alternating property, D(ei1, ei2, ..., ein) ≠ 0 only if i1, i2, ..., in are distinct. Since the n integers i1, i2, ..., in are chosen from 1, 2, ..., m, this can happen only if mn. In other words, if m < n, then D(A) = 0 and this generalization of determinant is trivial.

Now assume mn. Then D(A) will be nontrivial. Moreover, we may rearrange each choice (i1, i2, ..., in) into ascending order J = (j1, j2, ..., jn), 1 ≤ j1 < j2 < ... < jnm. In other words, we think of (i1, i2, ..., in) as a permutation of J and rewrite D(A) as follows (∑J is the sum over all subsets of {1, 2, ..., n} that contain m numbers)

D(A) = ∑Jall permutations (i1,i2,...,in) of J ai11ai22...ainn D(ei1, ei2, ..., ein).

Then by alternating property, we have

D(ei1, ei2, ..., ein) = sign(i1, i2, ..., in)D(ej1, ej2, ..., ejn)

where sign(i1, i2, ..., in) is computed by a sequence of transpositions that changes (i1, i2, ..., in) to ascending order. For example, the following sequence of transpositions

(6, 3, 8, 5) → (3, 6, 8, 5) → (3, 6, 5, 8) → (3, 5, 6, 8)

tells us sign(6, 3, 8, 5) = -1. Thus

D(A) = ∑J D(ej1, ej2, ..., ejn) ∑all permutations (i1,i2,...,in) of J sign(i1, i2, ..., in) ai11ai22...ainn.

Now ∑all permutations (i1,i2,...,in) of J sign(i1, i2, ..., in) ai11ai22...ainn is the same as the explicit formula for the determinant, except (1, 2, ..., n) is now replaced by J = (j1, j2, ..., jn). In other words, if we denote

A(J) = A(j1, j2, ..., jn) = n by n matrix formed by [row j1], [row j2], ..., [row jn] of A,

then

all permutations (i1,i2,...,in) of J sign(i1, i2, ..., in) ai11ai22...ainn = detA(J)

and we conclude the following.

Suppose D(A) is a multilinear and alternating function on columns of m by n matrices A. Then D(A) = 0 for m < n, and

D(A) = ∑1≤j1<j2<...<jnm D(ej1, ej2, ..., ejn) detA(j1, j2, ..., jn)

for mn.

Conversely, for arbitrarily chosen numbers DJ (one for each subset J of {1, 2, ..., n} that contain m numbers), the function D(A) = ∑J DJ detA(J) is a multilinear and alternating function. The total number of J's is n!/m!(n-m)!. In the special case m = n, the number is 1, and we recover this result.

Finally, the functions described in this section can be reformulated as linear functions over a suitable vector space ΛnRm, called exterior algebra, which is a fundamental tool in modern mathematics.