Matrix ^ ZZ -- power

Synopsis

i1 : R = ZZ/7[x]/(x^6-3*x-4)

o1 = R

o1 : QuotientRing

i2 : f = matrix{{x,x+1},{x-1,2*x}}

o2 = | x   x+1 |
     | x-1 2x  |

             2       2
o2 : Matrix R  <--- R

i3 : f^2

o3 = | 2x2-1  3x2+3x |
     | 3x2-3x -2x2-1 |

             2       2
o3 : Matrix R  <--- R

i4 : f^1000

o4 = | 3x5-2x4-2x2+2x+3 -2x5+x3-x2+x+1  |
     | x5+x4-2x2-3x+1   -x5+2x4-3x3+x-3 |

             2       2
o4 : Matrix R  <--- R

If the matrix is invertible, then f^-1 is the inverse.

i5 : M = matrix(QQ,{{1,2,3},{1,5,9},{8,3,1}})

o5 = | 1 2 3 |
     | 1 5 9 |
     | 8 3 1 |

              3        3
o5 : Matrix QQ  <--- QQ

i6 : det M

o6 = 9

o6 : QQ

i7 : M^-1

o7 = | -22/9 7/9   1/3  |
     | 71/9  -23/9 -2/3 |
     | -37/9 13/9  1/3  |

              3        3
o7 : Matrix QQ  <--- QQ

i8 : M^-1 * M

o8 = | 1 0 0 |
     | 0 1 0 |
     | 0 0 1 |

              3        3
o8 : Matrix QQ  <--- QQ

i9 : R = QQ[x]

o9 = R

o9 : PolynomialRing

i10 : N = matrix{{x^3,x+1},{x^2-x+1,1}}

o10 = | x3     x+1 |
      | x2-x+1 1   |

              2       2
o10 : Matrix R  <--- R

i11 : det N

o11 = -1

o11 : R

i12 : N^-1

o12 = {3} | -1     x+1 |
      {1} | x2-x+1 -x3 |

              2       2
o12 : Matrix R  <--- R

i13 : N^-1 * N

o13 = {3} | 1 0 |
      {1} | 0 1 |

              2       2
o13 : Matrix R  <--- R