Application of Cayley-Hamilton Theorem 

Application of Cayley-Hamilton Theorem
  -20  12 
61  -168  108 
5 529 -1208 732
4741  -10568  6348 
42649  -94808  56892 
383821  -852968  511788 
3454369  -7676408  4605852 
10  31089301  -69087368  41452428 
11  279803689  -621786008  373071612 
12  2518233181  -5596073768  3357644268 

If the characteristic polynomial of the matrix A is p(x) = x3 - 9x2 + 20x - 12, then A5 = 529A2 - 1208A + 732I.

See also: Calculation of An from an Annililating Polynomial