MATH 553-C MIDTERM EXAMINATION 3 Name GT Email @gatech.edu Please read all instructions carefully before beginning. Please leave your GT ID card on your desk until your TA scans your exam. Each problem is worth 0 points. The maximum score on this exam is 50 points. You have 50 minutes to complete this exam. There are no aids of any kind (notes, text, calculator, etc.) allowed. Please show your work unless instructed otherwise. You may cite any theorem proved in class or in the sections we covered in the text. Good luck!
Problem. [2 points each] In the following questions, if the statement is true, circle T; if it is always false, circle F. All matrices are assumed to have real entries. a) T F Every 2 2 matrix with eigenvalues 0 and is diagonalizable. b) T F Every 3 3 matrix with eigenvalues 0 and is invertible. c) T F If A is a square matrix and det(a) 0, then the rows of A are linearly independent. d) T F Every upper-triangular square matrix is diagonalizable. e) T F Every upper-triangular square matrix does not have any complex eigenvalues. a) True: it has two distinct eigenvalues, so the two corresponding eigenvectors are linearly independent. b) False: a matrix is invertible if and only if zero is not an eigenvalue. c) True: det(a) 0 implies det(a T ) 0, which implies the columns of A T (which are the rows of A) are linearly independent by the invertible matrix theorem. d) False: any diagonal matrix is upper-triangular, but a shear is upper-triangular and not diagonalizable. e) True: the eigenvalues are the entries on the main diagonal.
Problem 2. [2 points each] In this problem, it is not necessary to show your work or justify your answers. a) Give an example of a 2 2 matrix which is neither invertible nor diagonalizable. b) Which of the following 3 3 matrices with real entries are necessarily diagonalizable? Circle all that apply.. A matrix with three distinct real eigenvalues. 2. A matrix with three distinct real eigenvectors. 3. A matrix with eigenvalues 2 and 3 such that λ = 3 has geometric multiplicity equal to 2. 4. A matrix with three linearly independent eigenvectors. c) Write a correct definition of an eigenvector: v is an eigenvector of an n n matrix A provided that d) What is the area of the parallelogram in the picture? (The grid marks are spaced one unit apart.) e) Let A be a 4 4 matrix with determinant 3. What is det( A)? a) 0 0 0 b). This is diagonalizable. 2. Every 3 3 matrix has at least three eigenvectors, but they may be multiples of each other. 3. This is diagonalizable: the algebraic and geometric multiplicity of 2 must be one. 4. This is diagonalizable. c)... v 0 and Av = λv for some scalar λ. d) The area is 3 det = 5. 2 e) det( A) = ( ) 4 det(a) = 3.
Problem 3. Consider the matrix A =. a) [3 points] Find the characteristic polynomial f (λ) of A. b) [3 points] Find all eigenvalues of A, real and complex. c) [4 points] For each eigenvalue, find a corresponding (real or complex) eigenvector. a) f (λ) = λ 2 Tr(A)λ + det(a) = λ 2 2λ + 2. b) We use the quadratic formula to find the eigenvalues: c) First we find a ( i)-eigenvector: i A ( i)i = i λ = 2 2 ± 4 8 = ± i. eigenvector trick v = i. An eigenvector with eigenvalue + i = i is just the complex conjugate v =. i
Problem 4. The matrix A = 3 4 7 2 5 7 2 4 6 has characteristic polynomial f (λ) = λ(λ ) 2 a) [4 points] Find bases for the 0-eigenspace and the -eigenspace. b) [2 points] What are the algebraic and geometric multiplicities of the eigenvalues? c) [3 points] Find a diagonal matrix D and an invertible matrix C such that A = C DC, or explain why A is not diagonalizable. d) [ point ] Write a nonzero vector in Nul A. a) The 0-eigenspace is the same as the null space, so we compute 0 x RREF PV F basis A 0 y = z 0 0 0 z For the -eigenspace, we compute 2 4 7 A I = 2 4 7 2 4 7 PV F x y z RREF = y 2 0 + z 2 7/2 0 0 0 0 0 0 7/2 0 basis 2 0., 7 0. 2 b) We see from (b) that 0 has algebraic and geometric multiplicity one, and has algebraic and geometric multiplicity 2. 2 7 0 0 0 c) C = 0, D = 0 0. 0 2 0 0 d) Any 0-eigenvector works, like.
Problem 5. A certain 2 2 matrix A has eigenectors v and w, pictured below, with corresponding eigenvalues 3/2 and /2. v w a) [3 points] Draw Av and Aw below. b) [4 points] Draw Ax and Ay below. y Av Aw Ay Ax c) [3 points] Compute A 00 (3v + 4w). (Your answer will be in terms of v and w.) 3 00 c) A 00 (3v + 4w) = 3A 00 v + 4A 00 w = 3 v + 4 00 w. 2 2 x