15. Suppose that the characteristic polynomial of some matrix A is found to be: p(a) = (λ— 4)³ ((A − 3)² (λ − 1) In each part, answer the question and explain your reasoning. a) What is the size of A? b) Is A invertible? c) How many eigenspaces does A have?

Given the following Matrix: A = 2 2 2 2 0 0 2 0 0 a) Identify the type of matrix (A) above? What does this mean about its diagonalizability? b) Compute the characteristic polynomial p(λ) of A. c) Find the eigenvalues of A. Order them so that λ1 ≥ λ2 ≥ λ3

4. Let A be a square matrix. (a) 1² is an eigenvalue of A? with the same eigenvector, v. Show that if A is an eigenvalue of A with corresponding eigenvector v, then (b) What is the characteristic polynomial of A2? Suppose that the characteristic polynomial of A is fa(1) = (A + 2)(A – 3). Suppose that v is an eigenvector of A². Is it necessarily true that v is also (c) an eigenvector of A? If so, then prove this fact; if not, then give an example that shows this is false.

in 2) Suppose that A is a square matrix and let p(x) be a polynomial with real coefficients with variable x. If p(x) = E a;x' i = 1 in , then define p (A) = E a¡A' . If, is an eigenvalue of A, the show that p (1) is an eigenvalue of p(A). i = 1

1. Let A = has eigenvalues X₁ = 2 2 3 -1 -2 2 1 and X₂ 5. You can assume, and do NOT need to show, that A Compute algebraic and geometric multiplicities of X₁ and ₂.

3. Consider the following matrix and answer the following questions: -CO A = (a) 12 points Without doing any calculations explain why A is diagonalizable. A is diagnozible because it's olso is invertible Compute the eigenvalues of A. (5² (c) 16 points) Compute the eigenvectors of A. -1- AC

Consider the following statements: 1. If A is a square matrix and Av = λv, for some nonzero number λ, then v is an eigenvector of A. 2. If λ is an eigenvalue of a square matrix A of order n, then the homogeneous linear system (A−λIn)x = 0 has infinite solutions. 3. The characteristic polynomial p(λ) of a square matrix A, of order n, has degree n and its real roots are eigenvalues ​​of A. 4. If λ is an eigenvalue of a square matrix A, then the set of eigenvectors of A associated with λ is a vector space.   Choose an option: (a) Statements 2 and 3 are true (b) Statements 1, 2, 3 and 4 are false. (c) Statements 1, 2 and 3 are true. (d) Only statements 2 and 4 are false. (e) All statements are true.

For the following matrix: a) find the eigenvalues and the eigenvectors b) check the answer by using the orthogonal property (x(i))T(x(j))=1(i≠l) c) determine the three normalized eigenvectors by using (x(i))T(x(i))=1(i≠l)