Elementary Linear Algebra Review for Exam 3 Exam is Friday, December th from :5-3:5 The exam will cover sections: 6., 6.2, 7. 7.4, and the class notes on dynamical systems. You absolutely must be able to do each of the following. Section 7. Determine whether or not a given vector is an eigenvector of a given matrix (To see if x is an eigenvector of A we compute Ax. If x and Ax are multiplies of each other and x 0, then x is an eigenvector. Otherwise x is not an eigenvector.) Determine whether or not a given number is an eigenvalue of a given matrix (To see if k is an eigenvalue of A we check the singularity of A ki. If A ki is singular, k is an eigenvalue. Otherwise, k is not an eigenvalue.) Use the definitions of eigenvector and eigenvalue for a matrix to find eigenvectors and eigenvalues for a related matrix. Determine a basis for an eigenspace of a matrix. Be able to explain and use the fact that eigenvectors corresponding to distinct eigenvalues are linearly independent. Section 7.2 Compute the characteristic polynomial of a matrix (at most 4 by 4). Determine the eigenvalues of a matrix. Explain why k is an eigenvalue of A if and only if c A (k) = 0, where c A is the characteristic polynomial of A. Section 7.3 Know and use what it means for a matrix to be diagonalizable. Be able to explain why if S is an invertible matrix and K is a diagonal matrix such that AS = SK, then the columns of S are eigenvectors of A. and the diagonal entries of K are the corresponding eigenvalues. Decide if a given matrix is diagonalizable or not, and if it is to determine S and K. Use the formulas SKS and S AS = K compute powers or other complicated expressions involving A. Dynamical systems Set up the system of equations for a dynamical system. Determine the transition matrix Use the transition matrix and eigenpairs to compute x(t) for specific t Use the transition matrix and eigenpairs to describe the limiting behavior. Sections 6. and 6.2. Calculate the determinant using Laplace expansion along a row or column.
Calculate the determinant using row reduction. (Remember the determinant of A equals the determinant of RREF(A) divided by the product of the fudge factors. Each time you interchange two rows you get a fudge factor of, each time multiply a row by k you get a fudge factor of k. Calculate a determinant using the pattern method. Use properties of the determinant such as det(ab) = det(a) det(b) and det(a) = 0 if and only if A is not invertible. Class notes Set-up the transition matrix for a population model. Use the transition matrix and the initial data to determine the population distribution at a given time (e.g. t =, 2, 3). Find a steady-state vector, and use it to find the limiting distribution, and interpret the long-term behavior of the system. Practice Problems. An biologist is testing the olfactory senses of mice, and runs the following experiment. A Christmas Party is held in a house whose floor plan is illustrated below. The caterer, from experience, knows that every 5 minutes the guests will are equally likely to move to one of the adjacent rooms or to stay in their current location. (a) Describe A so that x(t + ) = Ax(t) models this system. What do the components of A represent? (b) Find the steady-state vector for A. (c) What, in every day language, do the entries of the steady-state vector mean? (d) How should the caterer distribute the 60 trays of food that have been prepared (that is, how many trays should be put in each room)? Justify your answer. 2. The Planning Commission of a county has determined that it would be ideal to have a 2 to ratio between urban and rural populations in the county (that is, for every 2 urban citizens there should be rural citizen). They know that each year typically /4 of the rural citizens move in to the city. What fraction p of the urban citizens should they allow to move into the rural areas each year, in order to have a 2 to ratio in the long run? Your answer should include setting up a transition matrix that involves p, identifying a steady- state vector, and then finding p. 3. Let A be an n by n matrix. Then an eigenvector of A is, and an eigenvalue of A is. 4. Let 0 0 2 2 2 0 x = 0, and y = 0 (a) Is x an eigenvector of A? (b) Is y an eigenvector of A?
(c) Is 2 an eigenvalue of A? If so, find a corresponding eigenvector. (d) It is known that is an eigenvalue of A. Find a basis for the corresponding eigenspace. 5. Let where a is a real number. a 2 0 2 0, (a) For which value of a is an eigenvector of A? (b) For which value of a is 0 an eigenvalue of A? 0 6. (a) Show that if x is an eigenvector of the matrix A corresponding to the eigenvalue 2, then x is an eigenvector of A 3 + I. What is the corresponding eigenvalue? (b) Show that if y is an eigenvector of the matrix A corresponding to the eigenvalue 3 and A is invertible, then y is an eigenvector fo A. What is the corresponding eigenvalue? (c) Suppose that P is an invertible matrix, and R and S are matrices with P RP = S. Show that if z is an eigenvector of S with corresponding eigenvalue 7, then P z is an eigenvector of R. What is the corresponding eigenvalue? 7. The matrix A is 4 by 4, and the nonzero vectors v, v 2, v 3, v 4 satisfy Av = 2v, Av 2 = 2v 2, Av 3 = v 3 and Av 4 = 7v 4. (a) Explain why v 2, v 3, and v 4 are linearly independent. (b) Must v, v 2, v 3 and v 4 be linearly independent? Explain. 8. Find the characteristic polynomial of the following matrix. 2. 0 3 0 9. (a) Find the characteristic polynomial for the matrix (b) Find the eigenvalues of A. 0 0 0 0 0 2 3 (c) For each of the eigenvalues found in (b) determine all of the corresponding eigenvectors. 0. Let c be a constant and let c 0 c 2. (a) Carefully explain why 3 is an eigenvalue of A exactly when det(a 3I) = 0. (b) Compute det(a 3I), and use it to decide for which value(s) of c we have that 3 is an eigenvalue of A.
. Let 0 3 2 3 0 The characteristic polynomial of A equals x 3 + 4x 2 4x. (a) Find the eigenvalues of A. (b) For each eigenvalue of A find a basis for the corresponding eigenspace of A. Is A diagonalizable? If so, find an invertible matrix S and a diagonal matrix K such that S AS = K. 2. Below the matrix A is factored in the form SKS. 2 0 0 2 0 0 0 /2 0 0 0 0 0 /2 (a) Find the eigenvalues of A 2 4 0 0 0 2 (b) For each eigenvalue of A find a basis for the corresponding eigenspace. (c) Find lim t A t.. 3. Let 7 7 s 2 s 3 4. (a) The eigenvalues of A are and 2. For which value of s is A diagonalizable? (b) For this value of s find an invertible matrix S and a diagonal matrix K such that AS = SK. (c) Find a different invertible matrix P, and a different diagonal matrix K such that AS = SK. 4. Determine which of the following matrices are diagonalizable. For those that are diagonalizable, find a P and K such that AS = SK. (a) (b) (c) 0 0 2 0 3 0 0 0 0 0 2 3 4 0 0 0 2 5. It is known that A is a 3 by 3 matrix such that AS = SK where S = 2 3 and D = 0 0 0.8 0. 4 9 0 0.7 (a) Determine the eigenvalues of A, and give corresponding eigenvectors for each of the eigenvalues. (b) Let v, v 2, v 3 be the eigenvectors you found in (a). Let u be any vector in R 3. Why are there constants c, c 2, and c 3 such that u = c v + c 2 v 2 + c 3 v 3? (You shouldn t have to do any row reduction here.)
(c) Suppose now that u = (/3)v (/4)v 2 + (/3)v 3. Express Au as a linear combination of v, v 2 and v 3. (d) Express A 2 u as a linear combination of v, v 2 and v 3. (e) For any positive integer t, express A t u as a linear combination of v, v 2, v 3. (f) Determine lim t A t u. (g) If A represents the transition matrix for a dynamical system, how does one interpret the answer to (f)? 6. There are two types of Furbies, juvenile and adult. Each year /2 of the juvenile Furbies survive to adulthood, 2/3 of the adult Furbies live, and on average each adult Furbie produces 6/9 baby Furbies. (a) Explain how the above description leads to the matrix [ ] 0 6/9 M = /2 2/3. (b) Use the characteristic polynomial of M to show that the eigenvalues of M are 4/3 and 2/3. (c) Find eigenvectors u and v for M corresponding to 4/3 and 2/3. (d) Initially there are 00 juvenile Furbies and 20 adult Furbies. So x(0) =. (e) Explain why there are constants c and d so that x(0) = cu + dv. (f) Use (e) to express x(t) in terms of c, d, u and v. (g) Based on you answer to (f), what will happen to the Furby population in the long run? What will the ratio of juvenile to adult Furby s be in the long run? 7. Adult Furbies have become a delicacy, and to prevent the extinction of Furbies you ve been asked to determine the largest fraction h of adult Furbies that can be harvested each year. (a) Explain why the transition matrix when there is harvesting becomes [ ] 0 6/9 N = /2 2/3 h (b) Explain what happens to the population of Furbies if N has an eigenvalue greater than. (c) Explain what happens to the population of Furbies if the absolute value of each eigenvalue is less than. (d) Find h so that the matrix N has eigenvalue. (e) Why is the h found in (d) the largest fraction of adult Furbies that can be harvested each year (without the population going extinct)? 8. (a) Use the recursive method to compute the determinant of the matrix 0 0 0 0 0 2 0 0 0 0 3. 4 0 0 0 (b) Use the row-reduction method to compute the determinant of 0 2 3 4 0 2 3. 0 0 0 4 (c) Why is the matrix in (b) invertible? What is the determinant of its inverse?
9. Let a b c d e f g h i Assume that det(a) = 3, and det(b) = 5. and B = r s t u v w x y z (a) Using A, find a matrix C whose determinant is 3. (b) Using B, find a matrix D, other than B, whose determinant is 5. (c) Using A and B, find a matrix E whose determinant is 75. 20. I started with a 3 by 3 matrix A and did the following row operations: () Switch rows and 3 (2) Added 4 times row to row 2 (3) Added 3 times row one to row 3 (4) Multiplied row 2 by /2. The matrix I obtained is: 2 3 0 4 0 0 5 (a) What is det(a)? (b) Is A invertible, why or why not? 2. Calculate the determinants of each of the following matrices, and determine which are invertible. [ ] 0 2 3 2 3 2, B =, C = 2 3 4 3 4 0 0 0. 3 5 7 3 5 8 22. Use the pattern method to compute the determinant of the following matrix 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 23. Construct a 5 by 5 matrix using the numbers ±2, 000 whose determinant is negative. Explain why the determinant is negative, without computing the determinant. (Hint: think patterns!)..