Given the matrix P which is a 3x3 matrix so the characteristic equation | P-ßI |=0 must have 3 eigen values and for each eigen value we would have infinite eigen vectors corresponding to the eigen value.
Let us suppose the 1 remaining eigen value is ×. Thus the eigen values would be x, 2+i, 3 . Now we also know that if a system has a complex variable as a solution, then it implies that the conzugate of the complex variable must also exist. Hence x=2-i.
Thus the eigen values : 2-i, 2+i, 3.
Determinant : Product of the eigen values : (2-i)*(2+i)*3 = 15