The polynomial in an indeterminate given by evaluation of the determinant is called the characteristic polynomial of . It is a monic polynomial of degree n. Therefore the polynomial equation has at most n different solutions, i.e., eigenvalues of the matrix. They may be complex even if the entries of are real. According to the Cayley–Hamilton theorem, , that is, the result of substituting the matrix itself into its own characteristic polynomial yields the zero matrix.
The polynomial in an indeterminate given by evaluation of the determinant is called the characteristic polynomial of  It is a monic polynomial of degree n Therefore the polynomial equation has at most n different solutions ie eigenvalues of the matrix They may be complex even if the entries of are real According to the Cayley–Hamilton theorem  that is the result of substituting the matrix itself into its own characteristic polynomial yields the zero matrix