Mathematical Methods for Engineers 1 (AMS10/10A)

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1 Mathematical Methods for Engineers 1 (AMS10/10A) Quiz 5 - Friday May 27th (2016) 2:00-3:10 PM AMS 10 AMS 10A Name: Student ID: Multiple Choice Questions (3 points each; only one correct answer per question) 1. Let A,B M 3 3 (R), and suppose that det(a) = 2 and det(b) = 3. Then det(2ab T ) equals [ ] The matrix A = has eigenvalues λ 1 = 0 and λ 2 = 6. Which of the following is an eigenvector 1 1 of A[ corresponding ] [ to] λ 1 = 0? [ ] [ ] Consider a matrix A M n n (R) (n 2) and suppose that rank(a) = n 1. Then null(a) = {0 R n} det(a) = n 1 At least one of the eigenvalues of A is zero A is invertible 4. Suppose that a 5 5 matrix A has an eigenvalue λ with algebraic multiplicity 3 and that rank(a λi) = 4. The dimension of the eigenspace of A associated with λ is The determinant of the matrix A = is The eigenvalues of the matrix A = are {0} Can not be determined {0,3, 1} {1,7, 1} 7. Suppose that a full rank matrix A has an eigenvalue λ with algebraic multiplicity 4. Let V λ be the eigenspace of A associated with λ. Then, dim(v λ ) = 6 dim(v λ ) = 0 dim(v λ ) 4 dim(v λ ) > 4 8. Consider two similar matrices A and B, and let P be the similarity transformation such that A = PBP 1. Which one of the following statements is always true? A and B have the same eigenvectors A is diagonalizable A and B have the same eigenvalues A is invertible 1

2 9. Let λ be an eigenvalue of a matrix A. The eigenspace of A corresponding to λ is a subspace of the column space of A λi the nullspace of A λi the null element R n a subspace of the nullspace of A 10. Eigenvectors of a matrix A M n n (R) corresponding to distinct eigenvalues can be linearly dependent can be zero are always linearly independent always form a basis of R n True or False Questions (20 points) Identify whether the following statements are true or false. If a statement is true, justify it. If false, provide a simple counterexample or explain why you think the statement is false. 1. All diagonalizable matrices are invertible (Hint: a diagonal matrix is diagonalizable, is it always invertible?) 2. If λ = 0 is an eigenvalue of a matrix A then det(a) = 0 3. If v is an eigenvector of A, then v is an eigenvector of A k for any k N, i.e., k = 1,2,3, A matrix A M n n (R) is diagonalizable if and only if it has n distinct eigenvalues. 2

3 Exercise 1 (10 points) Consider the matrix A = 1. Find all eigenvalues and eigenvectors. [ Show that A k = A for any odd k, i.e., k = 1,3,5,7,... (Hint: use the fact that A is similar to a diagonal matrix D, i.e., A = PDP 1 ). ]. Exercise 2 (10 points) Let Q M n n (R) be an orthogonal matrix, i.e., a matrix that satisfies QQ T = I. Show that the determinant of Q is either 1 or -1. 3

4 7 4 0 Exercise 3 (15 points) Consider the matrix A = Find the eigenvalues of A along with their algebraic and geometric multiplicity. 2. Determine the basis of the eigenspace corresponding to each eigenvalue. 3. Determine whether A is diagonalizable. If so, give P and D such that A = PDP 1 (there is no need to show A = PDP 1, just state P and D ). 4

5 Extra Credit (10 points) Let A M n n (R) be invertible and diagonalizable. Show that if {λ 1,...,λ n } are eigenvalues of A then {1/λ 1,...,1/λ n } are eigenvalues of A 1. (Hint: use the similarity transformation between A and the diagonal matrix D that has the eigenvalues of A along the diagonal). Extra Credit (10 points) Let A M n n (R). Show that if {λ 1,...,λ n } are eigenvalues of A then {λ 1 µ,...,λ n µ} are eigenvalues of A µi. 5

6 Exercise 4 (15 points) Consider a square matrix A. Write a Matlab or Octave function that computes the matrix B = n A k, (1) where n is an integer number and A k = AA A }{{}. For convenience, the function name, input and output k times are given below. k=1 function [B] = matrixseries(a,n) % Input A -> nxn matrix. % n -> number of terms in the series (1) % % Output B -> A + A*A + A*A*A +... end Note: Pseudo-code or other languages (e.g., C, Java, SPITBOL, PROLOG, COBOL, Simula-67, PIZZA, DYNAMO, Forth, KRYPTON, Pascal, IBM RPG II, etc.) will not receive any credit. 6