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EXAM Exam 1 Math 2360–102, Summer I, 2015 June 10, 2015 • Write all of your answers on separate sheets of paper. Do not write on the exam handout. You can keep the exam questions when you leave. You may leave when finished. • You must show enough work to justify your answers. Unless otherwise instructed, give exact answers, not √ approximations (e.g., 2, not 1.414). • This exam has 7 problems. There are 330 points total. Good luck! 60 pts. Problem 1. Consider the matrices " # " 1 −1 1 A= , B= 2 3 1 2 # " , 4 −1 0 2 −3 1 C= 5 # In each part, determine if the operation is defined and if it is defined, compute the answer. A. 2C B. A − B C. AB D. AC E. CA F. C T 50 pts. Problem 2. In each part you are given the augmented matrix of a system of linear equations, with the coefficient matrix in reduced row echelon form. Determine if the system is consistent and, if it is consistent, find all solutions. A. 1 0 0 0 B. 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 0 0 C. 1 0 0 0 0 1 0 0 −1 2 3 5 5 7 0 0 −1 4 −2 −1 3 1 0 0 0 0 1 0 1 2 1 5 0 40 pts. Problem 3. In each part you are given the augmented matrix of a system of linear equations, with the coefficient matrix in row echelon form. (Not reduced row echelon form!) Use back substitution to find all the solutions of the system. A. B. 1 0 0 2 1 0 −2 3 1 1 2 3 0 0 0 1 0 0 40 pts. −1 2 3 1 1 4 Problem 4. In each part, solve the linear system using the Gauss-Jordan method (i.e., reduce the coefficent matrix to Reduced Row Echelon Form). Show the augmented matrix you start with and the augmented matrix you finish with. It’s not necessary to show individual row operations, you can just find the Reduced Row Echelon Form with your calculator. A. 4x 2x 2x + 2y + 2y + 2y 2x x 3x + + + B. y y 2y 2 + z + z − z + z = 9 = 2 = 5 = = = 4 3 7 50 pts. Problem 5. In each part, use row operations to determine if the matrix A is invertible and, if so, to find the inverse. Display the inverse by itself. Give the matrix entries in fractional form. A. In this part, show the individual row operations and the resulting matrices. Be sure to display the label for the row operation being performed. You can use a calculator for the row operations, if you want. 1 2 0 A = 2 0 −1 0 2 0 B. In this part, it is not necessary to show the individual row operations (you can just use the rref key on the calculator). Show the augmented matrix you start with and the augmented matrix you finish with. 1 1 1 A = 0 1 2 1 2 3 60 pts. Problem 6. Consider row operations on matrices with 3 rows. Recall that for each row operation there is a corresponding 3 × 3 elementary matrix E so that EA is the same as the matrix obtained by applying the row operation to A. A. Consider the row operation R2 ↔ R3 . i.) Find the corresponding elementary matrix E. ii.) Find the inverse of E B. Consider the row operation R3 ← 5R3 . i.) Find the corresponding elementary matrix E. ii.) Find the inverse of E C. Consider the row operation R1 ← R1 − 3R3 i.) Find the corresponding elementary matrix E. ii.) Find the inverse of E. 3