box% sage
┌────────────────────────────────────────────────────────────────────┐
│ SageMath version 9.2, Release Date: 2020-10-24                     │
│ Using Python 3.8.6. Type "help()" for help.                        │
└────────────────────────────────────────────────────────────────────┘
sage: A = matrix([[0,0,-3],[1,0,6],[0,1,-3]])
sage: A
[ 0  0 -3]
[ 1  0  6]
[ 0  1 -3]
sage: A.characteristic_polynomial()
x^3 + 3*x^2 - 6*x + 3
sage: (1/A^2)
[   3    2    1]
[-7/3   -1    0]
[-2/3 -1/3    0]
sage: (1/A^2).characteristic_polynomial()
x^3 - 2*x^2 + 7/3*x - 1/9