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# exteriorPower(ZZ,Matrix) -- exterior power of a matrix

- Function: exteriorPower
- Usage:
    `exteriorPower(i,f)`
    `exteriorPower_i f`
- Inputs:
  - i, an integer
  - f, a matrix
- Optional inputs:
  - Strategy => ..., default value null, choose between Bareiss, Cofactor and Dynamic algorithms
- Outputs:
  - a matrix, the i-th exterior power of f.

## Description

    i1 : R = ZZ/2[x,y];

    i2 : f = random(R^3,R^{3:-1})

    o2 = | x   y   y   |
         | x+y y   x   |
         | x   x+y x+y |

                 3      3
    o2 : Matrix R  <-- R

    i3 : exteriorPower_2 f

    o3 = | y2       x2+xy+y2 xy+y2 |
         | x2       x2       0     |
         | x2+xy+y2 y2       x2+y2 |

                 3      3
    o3 : Matrix R  <-- R

The matrix may be a more general homomorphism of modules. For example,

    i4 : g = map(coker matrix {{x^2},{x*y},{y^2}}, R^3, id_(R^3))

    o4 = | 1 0 0 |
         | 0 1 0 |
         | 0 0 1 |

    o4 : Matrix

    i5 : g2 = exteriorPower(2,g)

    o5 = | 1 0 0 |
         | 0 1 0 |
         | 0 0 1 |

    o5 : Matrix

    i6 : target g2

    o6 = cokernel | xy x2 0  |
                  | y2 0  x2 |
                  | 0  y2 xy |

                                3
    o6 : R-module, quotient of R

## See also

- exteriorPower(ZZ,Module) -- exterior power of a module

## Ways to use this method:

- exteriorPower(ZZ,Matrix) -- exterior power of a matrix

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Macaulay2Doc/doc7.m2:244:0.