Preparation problems for the discussion sections on September 16th and 18th1. (1) Find a matrix E such that:ER1R2R3=R1R2− 2R1R3Which matrix E−1undoes the row operation implemented by E? What is E−1E?(2) Find a matrix F such that:FR1R2R3=R2R1R3Which matrix F−1undoes the row operation implemented by F ? What is F−1F ?(3) Find a matrix G such that:GR1R2R3=R13R2R3Which matrix G−1undoes the row operation implemented by G? What is G−1G?2. Consider the matrix:2 3 30 5 76 9 8Decompose the matrix A into LU , where L is a lower triangular matrix and U is an uppertriangular matrix. Then use this factorization to solve:2 3 30 5 76 9 8x1x2x3=225That means, find a vector c in R3such that:Lc =225and then find a vector x in R3such that:Ux = c3. Let A =2 −1 0 0−1 2 −1 00 −1 2 −10 0 −1 2, L =1 0 0 0−121 0 00 −231 00 0 −341, and U =2 −1 0 0032−1 00 043−10 0 054.(1) Show that A = LU.(2) Let Aibe the i × i matrix introduced by the first i rows and the first i columns of A, fori = 1, 2, 3. What is an LU decomposition of Ai, for i = 1, 2, 3?14. (more challenging) Let A and B be n × n matrices such that AB = I.(1) What is the reduced echelon form of A?(2) Show that BA = I.5. Answer the following true-false questions. Explain your answer.(1) If A is invertible then Ax = 0 has exactly one solution, x = 0.(2) If A is invertible then AB is also invertible.(3) If A and B are invertible then A + B is also invertible.(4) If A is invertible then the reduced echelon form of A is equal to I.6. If G =0 11 2, find G−1. Check that G−1G = I.7. Let A =2 1 24 2 12 1 1. Use the Gauss-Jordan method to either find the inverse of A or toshow that A is not invertible.8. Calculate the inverse of the matrix:2 1 0 −11 −1 1 00 0 0 11 1 1 19. Consider the equation:−d2udx2= 4π2sin 2πx, u(0) = u(1) = 0(1) Write down the 3 by 3 matrix equation with h =14.(2) Solve for u1, u2, u3and find their error in comparison with the true solution u = sin 2πxat x =14, x =12, and x
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