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midterm_F06 - MATH 202 MIDTERM Wednesday October 25...

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MATH 202, MIDTERM Wednesday, October 25, 7.30-9.00 PM Covers Chapters 1–5.1 of textbook Time: 1 hour 30 minutes Your name (print): . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Your instructor : . . . . . . . . . . . . . . . . . . . Time your class meets : . . . . . . Please show all work and explain your answers. Books, notes and calculators are not permitted on this exam. Write below and sign the Pledge: I pledge my honor that I have not violated the Honor Code during this examination.
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1. (14 points) (a) (6 points) Let A, B be two n × n matrices which satisfy AB +3 B +2 I = 0, where I is the n × n identity matrix. Show that matrix B is invertible. (b) (8 points) Let A be a 4 × 4 matrix such that rref ( A ) = 1 2 0 2 0 0 1 3 0 0 0 0 0 0 0 0 . (i) (4 points) Based on this information can you solve the system Ax = 0? If yes, find all solutions. (ii) (4 points) Based on this information can you solve the system Ax = b , where vector b = 1 - 1 0 0 ? If yes, find all solutions.
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2. (24 points) Let V be the plane in R 3 which is orthogonal to the line spanned by the vector v 1 = 1 - 1 1 and let P : R 3 R 3
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