3e-spring2011-exam_1_sample1

3e-spring2011-exam_1_sample1 - Math 3E Exam #1 Sample Laney...

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Unformatted text preview: Math 3E Exam #1 Sample Laney College, Fall 2006 Fred Bourgoin 1. Consider the following system of equations. y + 2 kz = 0 x + 2 y + 6 z = 2 kx + 2 z = 1 For what values of k does the system have no solutions, exactly one solution, and infinitely many solutions? Explain your answers. 2. Circle the matrix (or matrices) in reduced row echelon form. A = 1 1 0 1 0 0 1 0 0 0 0 2 , B = 1 0 0 0 0 1 0 1 0 0 1 2 0 0 0 0 , C = 1 2 0 0 0 0 0 0 0 0 1 0 , D = 1 2 0 0 0 0 1 0 0 0 0 1 3. Compute the products. (a) 1 1- 1- 5 1 1 1- 5 3 1 2 3 (b) 1- 1 1 1 1- 1- 2 1 2 3 3 2 1 2 1 3 4. Determine whether each of the given types of geometric transformations in R 2 is invertible. Briefly justify your answers. (a) Rotation (b) Reflection (c) Projection 1 5. Let x = [ 1 2 ] and y = [- 1 2 ] be vectors in R 2 . For each matrix below, identify the type of geometric transformation it represents (be precise!), and determine the images of x and y . (a) [- 1 1 ] (b) [ . 8 . 6 . 6- . 8 ] (c) [ . 5 0 . 5 . 5 0 . 5 ] 6. What is the rank of the matrix A = 1 2 0 2 3 0 0 1 3 2 0 0 1 4- 1 0 0 0 0 1 ? 7. Find a diagonal matrix D such that D 5 3- 9 = 2 1 . 8. Find the inverse of A = 2 5 0 0 1 3 0 0 0 0 1 2 0 0 2 5 . 9. If A is an invertible matrix and c is a nonzero scalar, is the matrix cA invertible? If so, what is the relationship between A 1 and ( cA ) 1 ? 10. The momentum P of a system of 2 particles in space with masses m 1 and m 2 , and velocities v 1 and v 2 is defined as P = m 1 v 1 + m 2 v 2 . Consider two elementary particles with velocities v 1 = 1 1 1 a nd v 2 = 4 7 10 . The particles collide. After the collision, their respective velocities are observed to be w 1 = 4 7 4 a nd w 2 = 2 3 8 ....
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This note was uploaded on 04/19/2011 for the course MTH 203 taught by Professor Staff during the Spring '08 term at Grand Valley State University.

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3e-spring2011-exam_1_sample1 - Math 3E Exam #1 Sample Laney...

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