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Linear Algebra-Homework 12 March 26, 2012 All problems are from the textbook, section 7.3 Note: we will need problem 3 and 4 later on. Problem 1 Find...

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Linear Algebra-Homework 12 March 26, 2012 All problems are from the textbook, section 7.2, 7.3 Note: we will need problem 3 and 4 later on. Problem 1 Find the eigenvalues and eigenvectors for the following matrices: 1. ± 6 3 2 7 ² 2. 1 1 0 0 2 2 0 0 3 3. 1 1 1 1 1 1 1 1 1 4. 3 0 - 2 - 7 0 4 4 0 - 3 5. 1 0 0 - 5 0 2 0 0 1 6. 1 1 0 0 - 1 - 1 2 2 0 7. 5 1 - 5 2 1 0 8 2 - 7 Problem 2 Consider the matrix A = ± 1 k 1 1 ² where k is an arbitrary real number. For what values of k will this matrix have two distinct real eigenvalues ? For what values of k will this matrix have no real eigenvalues ? Suppose both A and B are two n × n matrices. We say that A is similar to a B if there exists an invertible matrix P such that A = P - 1 .B.P . Problem 3 Show that similar matrices have the same characteristic polynomial and hence will have the same eigenvalues. 1
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Problem 4 Show that for a given n × n matrix A , both A and A T have the same characteristic polynomial and hence will have the same eigenvalues. Problem 5 Consider an n × n matrix A such that the sum of elements of each row equals 1. Show that 1 1 . . . 1 is an eigenvector for this matrix. Find the eigenvalue corresponding to this eigenvector. Problem 6 (OPTIONAL:) Problem 50 in section 7.2 is a very important exercise for math majors. It gives you a formula to find the factors and roots of a cubic polynomial. ———————————————————————————————————————————————————— Syllabus for second midterm: Chapter 5-Sections 5.1, 5.2, 5.3, 5.4. Chapter 6-Sections 6.1, 6.2, 6.3, Chapter 7- Sections 7.2, 7.3, 7.4 2
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