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quiz1_S06 - x 1-x 2 = 1 x 1 kx 2-4 x 3 = 3-x 1 2 x 2(1 k x...

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Princeton University Department of Mathematics Name: Instructor: Class time: MAT 202 – Linear Algebra with Applications QUIZ 1 – Spring 2006 This quiz is due Friday, March 3, at 2 pm at your instructor’s office. You have 60 minutes to complete your work. Please show all work and write neatly. Each of the five problems corresponds to 10 points. Books, notes, calculators, computers are not permitted on the quiz. As part of your obligations under the Honor Code, do not discuss this quiz with anyone until after the Friday 2 pm deadline. WRITE OUT AND SIGN THE PLEDGE: I pledge my honor that I have not violated the Honor Code during this examination. for the grading problem score 1 2 3 4 5 total
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2 (1) Solve the following system: x 1 + 2 x 2 + x 3 + x 4 = 1 2 x 1 + 2 x 2 - x 3 - x 4 = 0 x 1 - 2 x 2 - 5 x 3 + x 4 = 0 What is the rank of the corresponding coefficient matrix and what is the rank of the corresponding augmented matrix?
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3 (2) For what values of k does the following system:
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Unformatted text preview: x 1-x 2 = 1 x 1 + kx 2-4 x 3 = 3-x 1 + 2 x 2-(1 + k ) x 3 = 0 have (a) infinitely many solutions? (b) no solutions? (c) exactly one solution? Justify your answers and provide a brief geometric interpretation. 4 (3) Find the matrix A of the orthogonal projection onto the plane x 1-x 2 + x 3 = 0 . Describe geometrically the transformation given by A 2 . Is A invertible? 5 (4) Find the inverse of the matrix A = 1 1 4 1 2 3 1 6 (5) Determine whether the following statements are true or false. No explanations, no credit. (a) Given two square matrices A and B , if AB is invertible, then so is A . (b) The composition of a shear, a reflection and a rotation in the plane is invertible. (c) There exist matrices A of size 3 × 2 and B of size 2 × 3 such that AB = I 3 is the identity matrix....
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