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# com08fA - AMS Common Exam Part A June 2008 Name Part A Part...

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AMS Common Exam - Part A, June 2008 Name: ID Num. Part A: / 75 Part B: / 75 Total: / 150 This component of the exam (Part A) consists of two sections (Linear Algebra and Advanced Calculus) with four problems in each. Each question is worth 25 points; choose THREE questions to answer from EACH section. Each problem should be solvable in approximately 20 minutes or less. Provide your answer in the space provided, and show all work. If extra sheets are used, place them inside the booklet and note on the cover page how many additional pages are included. Good Luck!

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Section 1: Linear Algebra Choose three of the four problems to solve. 1. Consider the product of any two n -square matrices: C = AB (a) Prove that C is non-singular if and only if both A and B are non-singular. (b) Prove that the rank of C must be less than or equal to the smaller of the ranks of A and B . In other words: rank( C ) min(rank( A ) , rank( B ))
2. Consider the system of linear equations: x + pz = a x + qy + z = a rx + z = a (a) For what values of a, p, q, r R

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com08fA - AMS Common Exam Part A June 2008 Name Part A Part...

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