The Hamilton–Cayley theorem, or Cayley–Hamilton theorem, says that every $n\times n$ matrix is a zero of its own characteristic polynomial.
The ring of $n\times n$ matrices is not a field and in particular, if the polynomial is factored over the field of scalars (assuming the entries in the matrix are in a field) the matrix is in some instances not a zero of any of the factors.
Can anything interesting be said about the set of $n\times n$ matrices that are zeros of the characteristic polynomial?
Consider the Jordan normal form of any matrix. It is necessary and sufficient for all of the eigenvalues of a matrix to be eigenvalues of the underlying matrix and for the degree of the eigenvalue in the characteristic polynomial to be at least the size of the largest Jordan Block of that eigenvalue in the matrix.
First, powers of a matrix act independently on each Jordan block, so a matrix satisfies a polynomial iff all of its Jordan blocks satisfy it. It a given Jordan block satisfies a polynomial, then its entire diagonal is the value of the eigenvalue in the polynomial, so the eigenvalue of the matrix is a root of the polynomial. After subtracting the eigenvalue, we need for canonical nilpotent matrix of ones one above the diagonal to satisfy the polynomial. The value on the diagonal $i$ above the main diagonal is the coefficient of $A^i$ in the shifted polynomial, so the matrix satisfies the polynomial iff the eigenvalue had degree at least the size of the Jordan block.