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MATH 110 - Spring 2010 - Ribet - Midterm 2 (solution)

# MATH 110 - Spring 2010 - Ribet - Midterm 2 (solution) -...

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PROFESSOR KENNETH A. RIBET Last Midterm Examination March 18, 2010 2:10–3:30 PM, 10 Evans Hall Please put away all books, calculators, and other portable electronic devices—anything with an ON/OFF switch. You may refer to a single 2-sided sheet of notes. For numerical questions, show your work but do not worry about simplifying answers. For proofs, write your arguments in complete sentences that explain what you are doing. Remember that your paper becomes your only representative after the exam is over. Problem Possible points 1 6 points 2 12 points 3 6 points 4 6 points Total: 30 points 1. a. Use row operations to find the inverse of the matrix - 2 1 0 4 - 3 1 1 1 - 1 . I’m sure that all of you know how to do this and that most of you will do it correctly. The answer seems to be 2 1 1 5 2 2 7 3 2 . b. Let A be an m × n matrix of rank m and let B be an n × p matrix with rank n . Determine the rank of AB . Prove that your answer is correct. Think of L A : F n F m and L B : F p F n . Their ranks are equal to the dimensions of the spaces to which they are mapping. Thus these maps are onto . It follows that the same statement is true for L A L B = L AB .

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