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Determinant u . v  2 _ _ Dot Product

The phrases -/ . * and +/ . * are the determinant and permanent of square matrix arguments. More generally, the phrase u . v is defined in terms of a recursive expansion by minors along the first column, as discussed below.
 		   For vectors and matrices, the phrase x +/ . * y is equivalent to the dot, inner, or matrix product of math; other rank-0 verbs such as <. and *. are treated analogously. In general, u . v is defined by u@(v"(1+lv,_)) , restated in English below.
 

For example: 
   x=: 1 2 3 [ m=: >1 6 4;4 1 0;6 6 8
   det=: -/ . * 
   mp=: +/ . *
   x ([ ; ] ; det@] ; mp ; mp~ ; mp~@]) m
+-----+-----+----+--------+-------+--------+
|1 2 3|1 6 4|_112|27 26 28|25 6 42|49 36 36|
|     |4 1 0|    |        |       | 8 25 16|
|     |6 6 8|    |        |       |78 90 88|
+-----+-----+----+--------+-------+--------+
The monad u . v is defined as illustrated below:
   DET=: 2 : 'v/@,`({."1 u . v $:@minors)@.(0<{:@$) @ ,. "2'
     minors=: }."1 @ (1&([\.))

   -/ DET * m
_112

   -/ DET * 1 16 64
49

   -/ DET * i.3 0
1

   +/ DET * m
320

The definition u@(v"(1+lv,_)) given above for the dyadic case may be re-stated in words as follows: u is applied to the result of v on lists of “left argument cells” and the right argument in toto. The number of items in a list of left argument cells must agree with the number in the right argument. Thus, if v has ranks 2 3 and the shapes of x and y are 2 3 4 5 6 and 4 7 8 9 10 11, then there are 2 3 lists of left argument cells (each shaped 4 5 6); and if the shape of a result cell is sr, the overall shape is 2 3,sr .


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