Cayley–Hamilton Theorem

Every square matrix over a commutative ring satisfies its own characteristic polynomial: if p(λ) = det(λI - A) is the characteristic polynomial of an n×n matrix A, then p(A) = 0 (the zero matrix), expressing a polynomial identity fulfilled by A.

The statement that every square matrix over a commutative ring (in particular over a field) satisfies its own characteristic polynomial: if p(λ) = det(λI − A) then p(A)=0 when p is evaluated with matrix substitution.

The statement that every square matrix satisfies its own characteristic polynomial: if p(λ) = det(λI − A) is the characteristic polynomial of a square matrix A, then p(A) = 0 (the zero matrix).