1. Answer true or false.
(a) If A is a 3 × 3 matrix with spectrum σA = {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 AT have the same eigenvalues.
(d) If A is a 4 × 4 matrix with spectrum σA = {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 ρA(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:
![1685_img1.png](https://secure.tutorsglobe.com/CMSImages/1685_img1.png)
(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, σA, for matrix A.
(c) Find a basis for each eigenspace, N (A-λiI), 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 αn+1 = -αn + 6αn-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 αn+1 that only depends on n.
5. Consider the subspace U = span{u→1, u→2, u→3} of R4, where:
![1764_img2.png](https://secure.tutorsglobe.com/CMSImages/1764_img2.png)
(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:
![19_img3.png](https://secure.tutorsglobe.com/CMSImages/19_img3.png)
(a) Find the null space of AT.
(b) Use the Fredholm Alternative Theorem to determine whether the system is solvable for all b→ ∈ R4. Justify your reasoning.
7. Given the matrix:
![2109_img4.png](https://secure.tutorsglobe.com/CMSImages/2109_img4.png)
(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).