CR MATH 45 - Exam #1 Elimination and Matrix Operations (11 pages)

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Exam #1 Elimination and Matrix Operations



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Exam #1 Elimination and Matrix Operations

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Pages:
11
School:
College of the Redwoods
Course:
Math 45 - Linear Algebra
Linear Algebra Documents
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College of the Redwoods Mathematics Department Math 45 Linear Algebra Exam 1 Elimination and Matrix Operations David Arnold Copyright c 2000 David Arnold Eureka redwoods cc ca us Last Revision Date October 4 2002 Version 1 00 2 Essay Questions Read Carefully You have the weekend to complete the exam The exam is due on my desk at the beginning of class on Monday This exam is open notes open book All work must be done by hand but you may certainly use a calculator or computer to check your work where appropriate You must answer all of the exercises on your own You are not allowed to work in groups on the exam You are not allowed to enlist the aid of a tutor or friend to help with the exam You are not allowed to read the exercises in the exam then seek help on similar questions Once you open the exam and read the questions you may not seek any outside help of any kind From the moment you open the exam you must do everything by yourself Place the solution to each exercise on a separate sheet of paper On a good sheet of paper write out longhand and sign the following honor pledge I promise that all work found herein is my own I have received no help from tutors colleagues or other teachers I have honored all of the examination constraints listed in the directions Arrange the problems in order place these examination pages on top of your solutions then place the honor pledge on top of the examination as a cover sheet Staple Good luck Exercise 1 Use elimination and back substitution to solve the following system of equations All computations are to be performed by hand Show your work x1 2x2 3x3 12 2x1 3x2 x3 0 3x1 4x2 2x3 21 Exercise 2 The following matrix represents the reduced row echelon form of an augmented matrix for a system of three equations in unknowns x1 x2 and x3 1 0 2 3 0 1 3 4 0 0 0 0 What is the solution of the system Exercise 3 Let A be a matrix with three rows a What 3 3 matrix E adds 5 times row 2 to row 3 and then adds 2 times row 1 to row 2 when it multiplies matrix A b What 3 3 matrix F subtracts 2 times row 1 from row 2 and then subtracts 5 times row 2 from row 3 when it multiplies matrix A How is F related to matrix E in the previous question Exercise 4 Consider the coefficient matrix 1 2 1 3 0 A 2 4 5 2 For what triples b1 b b 2 b3 does the system Ax b have a solution 3 Exercise 5 Suppose that A is a 4 4 matrix such that the sum of the first two columns of matrix A equals the sum of columns 3 and 4 a Find a nonzero solution of Ax 0 where x is a column vector b Explain why matrix A is noninvertible singular Exercise 6 Suppose that B is a 5 5 matrix so that the sum of the first two rows of B is 3 times the sum of rows 3 4 and 5 Find a nonzero solution of yB 0 where y is a row vector Exercise 7 Show how you can express matrix 1 2 1 2 1 2 5 5 5 3 4 3 4 3 4 as the product of two vectors Exercise 8 Let A 1 5 0 2 a Find elementary matrices E and F so that EF A I b Write A 1 as a product of elementary matrices Exercise 9 You learned in class that you can find the inverse of a 2 2 matrix with the formula 1 1 d b a b c d ad bc c a Use this formula to find the inverse of the matrix cosh x sinh x sinh x cosh x Hint The hyperbolic cosine and sine are defined as follows cosh x Exercise 10 Given that ex e x 2 1 2 2 5 1 0 and sinh x 3 0 3 A 0 8 1 1 0 0 ex e x 2 0 1 0 find A 1 Note This problem is trivial if approached in the correct manner Look for an elegant solution Little credit will be given for crunch and grind solutions Exercise 11 For what value s of c is matrix A singular 1 2 A 1 5 2 6 not invertible 3 6 c 4 Exercise 12 Four by four matrix A consists of four 2 2 blocks I B A 0 I where I is the 2 2 identity matrix 0 is a 2 2 zero matrix all entries are zeros and B is any 2 2 invertible matrix a In block notation what is the inverse of matrix A b Craft a 4 4 matrix A that adheres to the pattern described above Use the technique developed in part a to find the inverse of your example c Use your matrix A from part b set up the augmented matrix A I and use row reduction to place your augmented matrix in the form I A 1 This result should agree with the solution found in part b Solutions to Exercises 5 Solutions to Exercises Exercise 1 Take the system x1 2x2 3x3 12 2x1 3x2 x3 0 3x1 4x2 2x3 21 and set up the augmented matrix 1 2 3 12 2 3 1 0 3 4 2 21 Subtract 2 times row 1 from row 2 Subtract 3 times row 1 from row 3 1 2 3 12 0 1 5 24 0 10 11 15 Subtract 10 times row 2 from row 3 1 2 3 12 0 1 5 24 9 0 39 225 This matrix represents the equivalent system x1 2x2 3x3 12 x2 5x3 25 39x3 225 We use back substitution to find the solution Solve 3 for x3 39x3 225 225 x3 39 75 x3 13 Substiute x3 75 13 in 2 and solve for x2 75 x2 5 24 13 375 312 x2 13 13 63 x2 13 Substitute x3 75 13 and x2 63 13 in 1 and solve for x1 63 75 x1 2 3 12 13 13 126 225 156 x1 13 13 13 99 156 x1 13 13 57 x1 13 1 2 3 Solutions to Exercises 6 Thus the solution is 57 13 63 x2 13 75 x3 13 x1 Exercise 1 Exercise 2 The augmented matrix 1 0 0 0 1 0 3 4 0 2 3 0 represents the system x1 2x2 3 x2 3x3 4 4 5 where x1 and x2 are pivot variables and x3 is a free variable Solve 4 and 5 for the pivot variables This gives solution x1 3 2x3 x2 4 3x3 x3 free Exercise 2 Exercise 3 a The elementary matrix 1 E32 0 0 adds 5 times row 2 to row 3 of matrix A when applied 1 E21 2 0 0 0 1 0 5 1 …


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