 With permutation matrices the determinant matches the signature, being +1 or −1 as the parity of the permutation is even or odd, for the determinant is an alternating function of the rows.Stronger than the determinant restriction is the fact that an orthogonal matrix can always be diagonalized over the complex numbers to exhibit a full set of eigenvalues, all of which must have (complex) modulus 1.The inverse of every orthogonal matrix is again orthogonal, as is the matrix product of two orthogonal matrices.