Final answer:
Both statements regarding the 6x6 matrix A are true . The characteristic polynomial will be a sixth-degree polynomial with real coefficients (option A), and A could have exactly one eigenvalue with algebraic multiplicity 6 (option B).
Step-by-step explanation:
The question deals with the properties of a 6x6 matrix A in terms of its characteristic polynomial and eigenvalues. Let's address each statement provided:
Statement A
The characteristic polynomial of A must be a polynomial of degree six with real coefficients. This statement is true. The characteristic polynomial of a square matrix is defined by the determinant of λI - A, where λ is a scalar and I is the identity matrix of the same size as A. Since A is a 6x6 matrix, the characteristic polynomial will indeed be a sixth-degree polynomial. Given that the matrix A has real entries, the coefficients of the characteristic polynomial will also be real.
Statement B
A could have exactly one eigenvalue, with algebraic multiplicity 6. This statement is true. A matrix can have a single eigenvalue repeated according to its algebraic multiplicity. The algebraic multiplicity is the number of times the eigenvalue appears as a root of the characteristic polynomial. Hence, it is possible for a 6x6 matrix to have one eigenvalue with algebraic multiplicity 6, which would mean that the eigenvalue is a repeated root of the characteristic polynomial.