1. Answer true or false.

(a) If A is a 3 × 3 matrix with spectrum σ_A = {1, 2, 3}, then A is guaranteed to be diagonalizable.

(b) If there exist a matrix P that diagonalizes A, then A is orthogonally diagonalizable.

(c) If A is any n × n matrix, then A and A^T have the same eigenvalues.

(d) If A is a 4 × 4 matrix with spectrum σ_A = {1, 2, 3, 3}, then it is possible that the dimension of N (A - 2I) is equal to 2.

(e) If λ is an eigenvalue of A, then the homogeneous  linear  system (A - λI)x^→ = 0^→ has   only the trivial solution.

(f) If the characteristic polynomial for the matrix A is given by ρ_A(λ) = (λ - 1)(λ - 3)²(λ + 2)³, then A must be invertible.

2. Prove:  If (λ, x^→) is an eigenpair for the matrix A, then (λ³, x^→) is an eigenpair for the matrix A³.

3. Given the matrix:

(a) Find the characteristic polynomial(equation) for the matrix A.

(b) Verify that the eigenvalues for   A are λ₁ = 5 and λ₂ = -1.  State the spectrum, σ_A, for matrix A.

(c) Find a basis for each eigenspace,  N (A-λ_iI), i = 1, 2.(Hint: Use Gaussian Elimination to solve.)

(d) The algebraic multiplicity of λ₁ = 5 is_____ and the geometric multiplicity of λ₁ = 5 is_____. 

(e) The algebraic multiplicity of λ₂ = -1 is _______ and the geometric multiplicity of λ₂ = -1  is____.

(f) Find the matrices P and Λ, where P  diagonalizes A.  (i.e.  P^-1AP = Λ)

4. Consider the recurrence relation given by α₀ = 0, α₁ = 1 and α_n+1 = -α_n + 6α_n-1.

(a) Rewrite the recurrence relation as a matrix equation,   w^→_n+1 = Aw^→_n.

(b) Find the characteristic equation, the eigenvalues and eigenvectors of the matrix found in part(a).

(c) Solve the recurrence relation, that is, give a formula for α_n+1 that only depends on n.

5. Consider the subspace U = span{u^→₁, u^→₂, u^→₃} of R⁴, where:

(a) Use the Gram-Schmidt Process to find an orthogonal basis, {v^→₁, v^→₂, v^→₃}, of U.

(b) Verify that this basis is orthogonal.  That is, show for each   i ≠ j that v^→_i ⊥ v^→_j

(c) Find an orthonormal basis,  {w^→₁, w^→₂, w^→₃}, for U . 

6. Given the system:

(a) Find the null space of A^T.

(b) Use the Fredholm Alternative Theorem to determine whether the system is solvable for all b^→ ∈ R⁴. Justify your reasoning.

7. Given the matrix:

(a) Find the eigenvalues for the matrix B.

(b) Find the eigenvectors of the matrix B corresponding to the eigenvalues you found in part (a).