  We exhibit explicit expressions, in terms of components, of discriminants,
determinants, characteristic polynomials and polynomial identities for matrices
of higher rank. We define permutation tensors and in term of them we construct
discriminants and the determinant as the discriminant of order $d$, where $d$
is the dimension of the matrix. The characteristic polynomials and the
Cayley--Hamilton theorem for higher rank matrices are obtained there from.
