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PROBLEM SET Assignment 1 Math 5316, Fall 2012 Sept 20, 2012 • Write all of your answers on separate sheets of paper. You can keep the question sheet. You can attach printouts of work done in Maple. • You must show enough work to justify your answers. Unless otherwise instructed, give exact answers, not √ approximations (e.g., 2, not 1.414). • This problem set has 8 problems. Good luck! Problem 1. In each part you are given the augmented matrix of a system of linear equations, with the coefficent matrix in reduced row echelon form. Determine if the system is consistent and, if it is consistent, find all solutions. A. B. C. 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 2 −5 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 2 −5 0 17 1 0 −2 3 0 0 0 1 −1 −1 0 −2 0 0 0 0 1 3 0 0 0 0 0 0 Problem 2. In each part you are given the augmented matrix of a system of linear equations, with the coefficient matrix in row echelon form (but not reduced row echelon form). Use backsubstitution to find all of the solutions. A. B. 1 2 0 0 1 0 −1 3 2 5 1 1 1 1 2 −1 0 0 0 0 1 2 0 0 0 0 0 0 5 1 3 2 1 1 0 0 Problem 3. In each part, you are given a system of linear equations. Form the augmented matrix of the system and perform row operations to 1 put the coefficient matrix in RREF. You can use Maple, but show the indvidual row operations (check yourself with Maple). Hint: When the left hand side is the same, you can do several systems at once by forming an augmented matrix with all of the right-hand sides to the right of the bar. A. 3 x1 + 22 x2 + 5 x3 = −55 3 x1 + 29 x2 + 6 x3 = −75 −4 x1 − 34 x2 − 6 x3 = 88 2 x1 + 20 x2 + 4 x3 = −52 B. 3 x1 + 22 x2 + 5 x3 = 174 3 x1 + 29 x2 + 6 x3 = 221 −4 x1 − 34 x2 − 6 x3 = −256 2 x1 + 20 x2 + 4 x3 = 152 C. 9 x1 + 21 x2 − 39 x3 + 6 x4 + 2 x5 = −36 2 x1 + 8 x2 − 12 x3 − 2 x4 + x5 = −13 2 x1 + 9 x2 − 13 x3 − 3 x4 + x5 = −15 5 x1 − 13 x2 + 3 x3 + 28 x4 − 2 x5 = 20 Problem 4. Consider the linear system whose augmented matrix is 1 2 1 2 1 1 b x 1 3 2 1 Determine the values of b for which the system has a unique solution. Find this solution (in terms of x). For the values of b such that the system does not have an unique solution, what values of x make the system consistent? Find all solutions of the system in this case. Problem 5. Consider the linear system 2 −1 A= 2 5 2 2 Ax = b where 3 1 1 0 1 1 0 2 −3 1 To determine the vectors b for which the system is consistent form the augmented matrix 2 3 1 b1 −1 1 0 b2 2 1 1 b3 5 0 2 b4 2 −3 1 b5 where the bi ’s are variables. Perfom row operations (you can use Maple) to reduce the coefficent matrix to Row Eschelon Form. Find equations that must be satisfied by the bi ’s for the system to be consistent. Give examples of nonzero vectors b so that the system is consistent, and is not consistent. This problem is easier in Maple, especially if you use the partial reduction procedure that can be downloaded from my website. Problem 6. Let H be the set of 3 × 3 1 H = 0 0 matrices of the form x y 1 z , 0 1 where x, y and z are real numbers. Note that the 3 × 3 identity matrix is in H. A. Find the formula for the product of two matrices in H. Show that H is closed under matrix multiplication. B. Using the previous part of the problem, solve a set of equations to find the inverse of a matrix in H. Show that H is closed under matrix inverse. C. Show that H is a group under matrix multiplication. This is the three dimensional Heisenberg Group. Is this group abelian? Problem 7. A matrix is said to be upper triangular if the entries below the main diagonal are zero. For example, 1 2 3 4 0 1 2 3 0 0 1 2 0 0 0 1 is upper triangular. A matrix A is upper triangular if and only if aij = 0 for i ? j. Show that the product of two upper triangular matrices is upper triangular. 3 Problem 8. Find two square matrices A and B so that A 6= 0, B 6= 0, A 6= B and AB = 0. Can A or B be invertible? 4