```rslt ← (f ##.alt g) mat                                 ⍝ Alternant.

Direct translation from the SAX 6.1 manual, from which I quote:

f.g ⍵ Alternant

The expressions -.×⍵ and +.×⍵ are for square matrix arguments ⍵, the determinant
and the permanent of mathematics.  The generalization to  arguments  other  than
plus, minus and times is based on construing  the  determinant as an alternating
sum (-⌿) over products over the diagonals of tables obtained  by  permuting  the
major cells of ⍵.

The model for the determinant described here is illuminating for what it reveals
operator (f.g).  The determinant is generated by the expression -.×⍵, where ⍵ is
a matrix of integers (usually square, but possibly with more rows than columns).
Only the top-left square portion of an array with more columns than rows is con-
sidered; that is, ⍵ is truncated to be (⌊/⍴⍵)↑⍵.

A model supplied by McDonnell and Hui for monadic inner product  is available in
the recursive function DOT shown below; the two functions f and g are assumed to
be subtract (⍺-⍵) and times (⍺×⍵) in the following example.

∇ z←DOT ⍵;M
   →(1≠¯1↑⍴⍵)/L0
   →0⊣z←f⌿⍵ ⍝ one-column case
  L0:M←~⍤1 0 ⊂ ⍳0@⍴⍵ ⍝ minors
   z←⍵[;⎕io] f.g DOT⍤2 ⍵[M;1↓⍳(⍴⍵)[1+⎕io]]
∇

For example, given the array M:

⊢M←3 3⍴6 7 2 1 5 9 8 3 4
6 7 2
1 5 9
8 3 4

the expression which DOT evaluates to produce the determinant uses  each  scalar
in the first column of M with subarrays called minors derived from the remaining
columns of the remaining rows:

(6×(5÷4)-3×9) - (1×(7×4)-3×2) - 8×(7×9)-5×2
360

Examples:

det ← -alt×                     ⍝ determinant

det 2 2⍴2 3 4 5
¯2
det 0⍪⍨ 2 2⍴2 3 4 5
¯2
det 2 2⍴0 1,1 0
¯1
det 1 1⍴3
3
det 1 0⍴3
0
det 0 1⍴3
1
det 0 0⍴3
1
hil ← {÷1+∘.+⍨(⍳⍵)-⎕io}         ⍝ Order-⍵ Hilbert matrix.

det hil 5
3.749295132E¯12

det 2 2⍴ 0 1, 1 0
¯1
,alt, ⎕←3 3⍴⎕A
ABC
DEF
GHI
┌───────────────┐
│AEIHFDBIHCGBFEC│
└───────────────┘

{⍺ ⍵}alt{⍺ ⍵} ⎕←3 3⍴⎕A
ABC
DEF
GHI
┌───────────────────────────────────────┐
│┌───────────┬─────────────────────────┐│
││┌─┬───────┐│┌───────────┬───────────┐││
│││A│┌──┬──┐│││┌─┬───────┐│┌─┬───────┐│││
│││ ││EI│HF│││││D│┌──┬──┐│││G│┌──┬──┐││││
│││ │└──┴──┘││││ ││BI│HC││││ ││BF│EC│││││
││└─┴───────┘│││ │└──┴──┘│││ │└──┴──┘││││
││           ││└─┴───────┘│└─┴───────┘│││
││           │└───────────┴───────────┘││
│└───────────┴─────────────────────────┘│
└───────────────────────────────────────┘
+alt+ 2 2⍴⍳4
10
+alt- 2 2⍴⍳4
¯2
+alt× ⎕←4 4⍴ 1 1 1 1, 2 1 0 0, 3 0 1 0, 4 0 0 1     ⍝ permanent
1 1 1 1
2 1 0 0
3 0 1 0
4 0 0 1
10