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exterior power of a matrix

Since the i-th exterior power is a functor, it applies to matrices as well as to modules.
i1 : R = ZZ[vars(0..19)]

o1 = R

o1 : PolynomialRing
i2 : ff = genericMatrix(R,4,5)

o2 = | a e i m q |
     | b f j n r |
     | c g k o s |
     | d h l p t |

             4      5
o2 : Matrix R  <-- R
i3 : exteriorPower (2,ff)

o3 = | -be+af -bi+aj -fi+ej -bm+an -fm+en -jm+in -bq+ar -fq+er -jq+ir -nq+mr
     | -ce+ag -ci+ak -gi+ek -cm+ao -gm+eo -km+io -cq+as -gq+es -kq+is -oq+ms
     | -cf+bg -cj+bk -gj+fk -cn+bo -gn+fo -kn+jo -cr+bs -gr+fs -kr+js -or+ns
     | -de+ah -di+al -hi+el -dm+ap -hm+ep -lm+ip -dq+at -hq+et -lq+it -pq+mt
     | -df+bh -dj+bl -hj+fl -dn+bp -hn+fp -ln+jp -dr+bt -hr+ft -lr+jt -pr+nt
     | -dg+ch -dk+cl -hk+gl -do+cp -ho+gp -lo+kp -ds+ct -hs+gt -ls+kt -ps+ot
     ------------------------------------------------------------------------
     |
     |
     |
     |
     |
     |

             6      10
o3 : Matrix R  <-- R
Note that each entry of in the above matrix is a 2 by 2 minor (the determinant of a 2 by 2 submatrix) of the matrix ff.

The signs are chosen so that this operation commutes with multiplication:
i4 : S = QQ[x_(1,1) .. x_(3,5), y_(1,1) .. y_(5,4)]

o4 = S

o4 : PolynomialRing
i5 : M = transpose genericMatrix(S,x_(1,1),5,3)

o5 = {-1} | x_(1,1) x_(1,2) x_(1,3) x_(1,4) x_(1,5) |
     {-1} | x_(2,1) x_(2,2) x_(2,3) x_(2,4) x_(2,5) |
     {-1} | x_(3,1) x_(3,2) x_(3,3) x_(3,4) x_(3,5) |

             3      5
o5 : Matrix S  <-- S
i6 : N = transpose genericMatrix(S,y_(1,1),4,5)

o6 = {-1} | y_(1,1) y_(1,2) y_(1,3) y_(1,4) |
     {-1} | y_(2,1) y_(2,2) y_(2,3) y_(2,4) |
     {-1} | y_(3,1) y_(3,2) y_(3,3) y_(3,4) |
     {-1} | y_(4,1) y_(4,2) y_(4,3) y_(4,4) |
     {-1} | y_(5,1) y_(5,2) y_(5,3) y_(5,4) |

             5      4
o6 : Matrix S  <-- S
i7 : exteriorPower(3,M*N) == exteriorPower(3,M) * exteriorPower(3,N)

o7 = true

See also