The characteristic polynomial of a 3×33×3 matrix can be factorised as (2−λ)2(5−λ)(2−𝜆)2(5−𝜆). Which of the following statements are true? Select all that apply.The geometric multiplicity of the eigenvalue λ=𝜆= 55 could be 22The number of linearly independent eigenvectors for the eigenvalue λ=𝜆= 55 must be 11The geometric multiplicity of the matrix cannot be greater than 22The geometric multiplicity of the eigenvalue λ=𝜆= 22 could be 22The algebraic multiplicity of the matrix is 22The algebraic multiplicity of the eigenvalue λ=𝜆= 22 cannot be determined from the information we are givenThe eigenvalue λ=𝜆= 22 could have infinitely many eigenvector directionsThe algebraic multiplicity of the eigenvalue λ=𝜆= 22 is 2

The characteristic polynomial of a 3x3 matrix is given as (2−λ)²(5−λ). This tells us that the eigenvalues of the matrix are 2 and 5. The exponent of (2−λ) and (5−λ) in the factorization of the characteristic polynomial gives us the algebraic multiplicity of the corresponding eigenvalue.

1. The geometric multiplicity of the eigenvalue λ=5 could be 2: False. The geometric multiplicity of an eigenvalue cannot exceed its algebraic multiplicity. Here, the algebraic multiplicity of λ=5 is 1, so its geometric multiplicity can only be 1.

2. The number of linearly independent eigenvectors for the eigenvalue λ=5 must be 1: True. The number of linearly independent eigenvectors for an eigenvalue is equal to its geometric multiplicity, which is 1 in this case.

3. The geometric multiplicity of the matrix cannot be greater than 2: True. The geometric multiplicity of a matrix is the maximum geometric multiplicity of its eigenvalues. Here, the maximum geometric multiplicity is 2 (for λ=2).

4. The geometric multiplicity of the eigenvalue λ=2 could be 2: True. The geometric multiplicity of an eigenvalue cannot exceed its algebraic multiplicity. Here, the algebraic multiplicity of λ=2 is 2, so its geometric multiplicity can be 1 or 2.

5. The algebraic multiplicity of the matrix is 2: False. The algebraic multiplicity of a matrix is the sum of the algebraic multiplicities of its eigenvalues. Here, the algebraic multiplicities of λ=2 and λ=5 are 2 and 1 respectively, so the algebraic multiplicity of the matrix is 2+1=3.

6. The algebraic multiplicity of the eigenvalue λ=2 cannot be determined from the information we are given: False. The algebraic multiplicity of λ=2 is given by the exponent of (2−λ) in the characteristic polynomial, which is 2.

7. The eigenvalue λ=2 could have infinitely many eigenvector directions: False. The number of eigenvector directions for an eigenvalue is equal to its geometric multiplicity, which can be at most 2 in this case.

8. The algebraic multiplicity of the eigenvalue λ=2 is 2: True. This is given by the exponent of (2−λ) in the characteristic polynomial.