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50 pts. Problem 1. 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. B. C. 1 0 0 0 0 0 −2 1 1 0 0 0 0 0 0 0 3 0 0 −5 −4 1 1 −1 0 0 0 0 0 0 1 0 0 0 0 0 −2 1 1 0 0 0 0 0 0 0 3 0 0 −5 −4 1 −1 1 0 0 1 0 0 0 40 pts. 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 3 1 5 0 0 Problem 2. Solve the linear system below. Use your calculator to find the RREF, but write down the augmented matrix and the matrix you wind up with, and then find all solutions. 2x1 + 4x2 + 2x3 = −6 x1 + x2 − x3 = −1 2x1 + x2 − 4x3 = 0 40 pts. Problem 3. Use row operations to determine if the matrix A is invertible and, if so, to find the inverse. Show the indivdual row operations, one by one. You can use a calculator to do the row operations if you wish. Give the matrix entries in fractional form. 2 6 A= 1 1 1 40 pts. Problem 4. Use Cramer’s rule to solve the following system. 2x1 + x3 = 1 x1 + 3x3 = 1 40 pts. Problem 5. Consider the matrix 1 A = −2 3 0 1 1 2 1 2 A. Find the cofactors A21 ,A22 and A23 . B. Show how to compute det(A) by using a cofactor expansion along some row or column. 40 pts. Problem 6. Use the method of elimination (i.e., row operations) to find the determinant of the matrix 5 4 A= 2 6 You can use a calculator to do the row operations, but show the row operations you apply and the intermediate matrices. Sorry, no credit for finding the determinant by another method! 2 EXAM Exam 1 Math 3351, Spring 2010 Feb. 21, 2011 • Write all of your answers on separate sheets of paper. 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 6 problems. There are 250 points total. Good luck!