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 
.
