Answer:
(A)three eigenvalues, all of them real.
(D)one real eigenvalue and two complex eigenvalues.
(G)only one eigenvalue -- a real one.
Step-by-step explanation:
Given an
matrix, the characteristic polynomial of the matrix is the degree n polynomial in one variable λ:
![p(\lambda) = det(\lambda I- A)](https://tex.z-dn.net/?f=p%28%5Clambda%29%20%3D%20det%28%5Clambda%20I-%20A%29)
If such
matrix A has real entries, its complex eigenvalues will always occur in complex conjugate pairs.
Therefore, for a
matrix with real entries, the following are possible:
(A)three eigenvalues, all of them real.
(D)one real eigenvalue and two complex eigenvalues.
(G)only one eigenvalue -- a real one.
A
matrix with real entries cannot have the following:
(B)three eigenvalues, all of them complex.
(C)two real eigenvalues and one complex eigenvalue.
(E)only two eigenvalues, both of them real.
(F)only two eigenvalues, both of them complex.
(H)only one eigenvalue -- a complex one.