Math 230 Final Exam

# Math 230 Final Exam - Honors Introductory Linear...

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Honors Introductory Linear Algebra (Math 230) – Spring 2011 Final Examination There are twelve problems and eight pages in this exam. Points (in parentheses) add to 105. You will be graded for 100 points. Provide appropriate justifications where required. Good luck! 1. (8) Find the characteristic polynomial and the eigenvalues of A where A = 2 3 5 4 1 6 0 0 3 . 2. (8) Let A = 1 3 1 - 3 0 0 0 1 - 3 4 0 0 0 - 1 2 - 2 - 6 - 2 6 0 . (a) Determine a basis for Col A . (b) Determine a basis for Nul A . (c) What is dim Nul A ? Explain. (d) What is rank A ? Explain. 3. (12) Let A and B be invertible n × n matrices, and let x be an eigenvector of the product AB corresponding to the nonzero eigenvalue λ . Show that the vector B x is an eigenvector of ( BA ) - 1 . What is the corresponding eigenvalue? Justify your steps. 4. (10) Let A + B and C be n × n invertible matrices. Solve the following equation for X . Justify each step in your solution.

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