A 3x3 matrix with real entries can have:. a) three eigenvalues, all of them real.
b) three eigenvalues, all of them complex.
c) two real eigenvalues and one complex eigenvalue.
d) one real eigenvalue and two complex eigenvalues.
e) only two eigenvalues, both of them real.
f) only two eigenvalues, both of them complex.
g) only one eigenvalue -- a real one.
h) only one eigenvalue -- a complex one.

a) three eigenvalues, all of them real.

d) one real eigenvalue and two complex eigenvalues.

Step-by-step explanation:

A  3x3 matrix with real entries can have : (a) three eigenvalues, all of them are real and (d) one real eigenvalue and two complex eigenvalues.

Let consider the equation for a  3x3 matrix with real entries :

From above ; we will notice that the polynomial is of 3°; as such there will be three eigenvalues in which all of them real.

Also ; complex values shows in pairs, a  3x3 matrix cannot have a three complex eigenvalues but one real eigenvalue and two complex eigenvalues.