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Math 2568, Exam #2, Part I, Summer 2015Name _________________________________Instructions: Show all work. You may not use a calculator on this portion of the exam. Give exact answers (yes, that means fractions, square roots and exponentials, and not decimals). Reduce as much as possible. Be sure to complete all parts of each question. Provide explanations where requested. When you are finished with this portion of exam, get Part II.Determine if each statement is True or False. For each of the questions, assume that A is n×n. (2 points each)TFA system that does have a unique solution cannot be solved with Cramer’s rule.TFA matrix is invertible if the determinant of the matrix is 0.TFIf λ is an eigenvalue of A, then λ2 is an eigenvalue of A2. TFIf zero is not an eigenvalue of A, then the determinant of A is non-zero.TFIf v is an eigenvector of A, then v is also an eigenvector of eA.TFIf A is invertible, then A is diagonalizable with real numbers.TFRow operations on a matrix do not change its eigenvalues.TFSimilar matrices always have the same eigenvectors.TFThe dimension of the eigenspace of an n×n matrix is always n.TFThe vector 11 is not an eigenvector of 5-278.TFA 7x7 matrix has four eigenvalues, two eigenspaces are one dimensional, two eigenspaces are two dimensional; therefore this matrix is diagonalizable. TFStochastic matrices, regardless of their size, always have at least one real eigenvector.TFThe real eigenvalues of a system of linear ODEs must always both attract or repel from the origin.TFIn a discrete dynamical system, the magnitude of λ determines whether a complex eigenvalues causes the origin to repel or attract. TFThe cross product is one type of inner product. TFNormalizing a vector refers to making a vector pointing in a particular direction have components that satisfy certain conditions.Find the determinant of the matrix 2543430-3021-430521-11200200 by the cofactor method. (12 points)Find the determinant of the matrix 1-2523-2401 by the row-reducing method. (11 points)Given that A and B are 3×3 matrices with det A = -2 and det B = 5, find the following. (4 points each)det ABd) det BT b) det A-1e) det 3Ac) det (-AB5)For each of the matrices shown below, find the eigenvalues and eigenvectors of the matrix. (7 points each)A=6-6-6-3B=-2-3-10Find the equilibrium vector of the matrix P=.8.3.2.7 algebraically. Be sure to properly normalize the vector. (8 points)For each of the situations below, determine the properties of the linear system of ODEs. Is the origin an attractor, a repeller, or a saddle point? Sketch the eigenvalues on the graphs provided (if they are real) and plot some sample trajectories. (5 points each)λ1=0.97, λ2=0.2, v1=12, v2=-13. 3838575508000λ1=75, λ2=-2, v1=21, v2=-23. 3838575508000λ1=12+72i, λ2=12-72i. 3838575508000For the vectors u=14-4, v=4-2-1, find the following: (4 points each)vA unit vector in the direction of u.u?vAre the two vectors orthogonal? Why or why not? ................
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