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# Nancy khalil spring 2018 homework 6 page 24 math 232

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Nancy Khalil, Spring 2018 Homework 6, page 2/4

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MATH 232 Applied Linear Algebra 2. Find the constants a , b , c , d , and e . 2.2 Eigenvalues and Eigenspaces Find all eigenvalues and corresponding eigenvectors of the matrix A = [ 1 3 3 1 ] 2.3 Eigenvalues and Determinants Let A ( λ ) = - λ 1 0 0 0 - λ 1 0 0 0 - λ 1 0 0 1 - λ 1. Show that | A ( λ ) | = λ 2 ( λ 2 - 1) 2. Find and describe the null space of A ( - 1). 3 Training questions Prove that the determinant of a 2 × 2 matrix is equal to the product of its eigenvalues. [Challenge] Prove that the determinant of an n × n matrix is equal to the prod- uct of its eigenvalues. (Hint: think about the factorization of the characteristic polynomial.) Nancy Khalil, Spring 2018 Homework 6, page 3/4
MATH 232 Applied Linear Algebra Legend (for “type” of question): RE = Routine Exercise: This is something you should be able to do in your sleep ;-). Your goal is to be able to answer these questions quickly and accurately every time. These form the foundations of your skill set. TC = Time Challenge: Speed and accuracy are important factors in solving this type of routine exercise. Try to do these exercises within the time limit, usually 5 minutes. If you need more time than that, it’s o.k., but keep practicing! Solving these routine exercises provides a foundation for solving more involved problems, and is essential in performing well on quizzes and exams. WP = Word Problem: Translating words into expressions (also known as modeling): Master this skill now, we will be using this all term.
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