Question

You only have 4 attempts at this problem. A real matrix has an even number of real eigenvalues. Choose:

a) There exists a real matrix with the eigenvalues λ and -λ.
b) A real eigenvalue of a real matrix always has at least one corresponding real eigenvector.
c) Every real matrix must have at least one real eigenvalue.

Answer 1

a) Yes, it is possible for a real matrix to have eigenvalues λ and -λ. b) No, a real eigenvalue of a real matrix may have a corresponding complex eigenvector. c) No, there exist real matrices without any real eigenvalues.

Step-by-step explanation:

a) There exists a real matrix with the eigenvalues λ and -λ.

Yes, it is possible for a real matrix to have eigenvalues λ and -λ. For example, consider the 2x2 matrix A = [0 -λ; λ 0]. The characteristic equation of A is λ^2 + λ^2 = 0, which has eigenvalues λ = i and λ = -i. Since the eigenvalues are complex conjugates, they have the form λ and -λ where λ is a real number.

b) A real eigenvalue of a real matrix always has at least one corresponding real eigenvector.

No, this is not always true. A real eigenvalue of a real matrix may have a corresponding complex eigenvector. For example, the 2x2 matrix B = [1 1; -1 1] has a real eigenvalue λ = 1, but its corresponding eigenvector is (1 + i, -1 + i), which is complex.

c) Every real matrix must have at least one real eigenvalue.

No, this is not true. There exist real matrices without any real eigenvalues. For example, the 2x2 matrix C = [0 -1; 1 0] has complex eigenvalues λ = i and λ = -i, but no real eigenvalues.

Answer 2

a) There exists a real matrix with the eigenvalues λ and -λ. b) A real eigenvalue of a real matrix always has at least one corresponding real eigenvector. c) Every real matrix must have at least one real eigenvalue.

Step-by-step explanation:

a) There exists a real matrix with the eigenvalues λ and -λ.

To determine if this statement is true, we need to understand that eigenvalues come in pairs for real matrices. If a real matrix has an eigenvalue λ, then it will also have an eigenvalue -λ. This is because the characteristic polynomial of a real matrix has real coefficients, and complex roots always occur in conjugate pairs. Therefore, option a) is correct.

b) A real eigenvalue of a real matrix always has at least one corresponding real eigenvector.

This statement is not always true. A real matrix can have real eigenvalues without any corresponding real eigenvectors. For example, consider the 2x2 matrix

(1 1)

(1 1)

The eigenvalues of this matrix are both 2, but there are no real eigenvectors.

c) Every real matrix must have at least one real eigenvalue.

This statement is not always true. A real matrix can have complex eigenvalues without any real eigenvalues. For example, consider the 2x2 matrix

(0 1)

(-1 0)

The eigenvalues of this matrix are complex numbers (i and -i), but there are no real eigenvalues.