Given
 |  |  | (1) |
 |  |  | (2) |
then
 | (3) |
where 
is the identity matrix. Cayley verified this identity for 
and 3 and postulated that it was true for all 
. For 
, direct verification gives
 |  |  | (4) |
 |  |  | (5) |
 |  |  | (6) |
 |  |  | (7) |
 |  |  | (8) |
 |  |  | (9) |
 |  |  | (10) |
 |  |  | (11) |
so
 | (12) |
The Cayley-Hamilton theorem states that an 
matrix 
is annihilated by its characteristic polynomial 
, which is monic of degree .