Explanation:

To prove that option B is correct, we need to show that every 5 x 5 matrix with real entries has at least one real eigenvalue.

Eigenvalues and Eigenvectors:
An eigenvalue of a matrix A is a scalar λ such that there exists a non-zero vector x satisfying the equation Ax = λx. The vector x is called the eigenvector corresponding to the eigenvalue λ.

Real Eigenvalues:
An eigenvalue is said to be real if it is a real number. In other words, the eigenvalue does not have an imaginary component.

Proof:

We will prove that every 5 x 5 matrix with real entries has at least one real eigenvalue by contradiction.

Assume that there exists a 5 x 5 matrix A with real entries that has no real eigenvalues.

Complex Conjugate Theorem:
If a polynomial with real coefficients has a complex root, then its conjugate is also a root.

Characteristic Polynomial:
The characteristic polynomial of a matrix A is given by det(A - λI), where det(A - λI) is the determinant of the matrix obtained by subtracting λ from the diagonal elements of A.

Complex Eigenvalues:
If a polynomial with real coefficients has complex roots, they must occur in conjugate pairs. Therefore, if a 5 x 5 matrix A has no real eigenvalues, all of its eigenvalues must be complex.

Complex Eigenvalues of Real Matrix:
If A is a real matrix, then the complex eigenvalues must occur in conjugate pairs. However, since A has no real eigenvalues, there must be an odd number of complex eigenvalues.

Odd Number of Complex Eigenvalues:
Let λ_1, λ_2, ..., λ_k be the complex eigenvalues of A. Since there is an odd number of complex eigenvalues, there must exist at least one complex eigenvalue λ_i such that its conjugate λ_i* is not an eigenvalue.

Complex Conjugate Theorem:
By the complex conjugate theorem, if λ_i is an eigenvalue, then its conjugate λ_i* must also be an eigenvalue. But we have assumed that λ_i* is not an eigenvalue. This contradiction implies that our assumption that A has no real eigenvalues is false.

Therefore, every 5 x 5 matrix with real entries must have at least one real eigenvalue. Hence, option B is correct.