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Step 1: Find the singular value decomposition of matrix A. Show more…
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The TMA covers only chapters 1 and 2. It consists of four questions; each question is worth 10 marks. Please solve each question in the space provided. You should give the details of your solutions and not just the final results. Q-1: [5×2 marks] Answer each of the following as True or False justifying your answers: a) If A and B are 5 × 5 matrices such that AB = -BA, then AB is singular. b) If (AB)^2 = A^2B^2, then AB = BA. c) If A is an n × n matrix such that A^2 = 0, then (I - A)^-1 = I + A. d) |a1 a2 a3| |c1+2a1 b1 a1| |b1 b3 b3| = |c3+2a3 b3 a3| |c1 c2 c3| |c2+2a2 b2 a2| e) The vector (1,2,0,4) is a linear combination of the vectors (1,0,1,2), (2,0,2,3) and (3,0,2,2). Q-2: [6+4 marks] Let A = [1 0 1] and B = [1 2 3] [1 1 2] [2 3 0] [2 0 1] [3 0 5] a) Find A^-1 B^T, if exists. b) Find the matrix C such that (AC + B)^T = A^T + B. Q-3: [5+5 marks] a) Solve the linear system { x1 + 4x4 = 4 -2x1 + x2 - x3 - 3x4 = -9 -2x1 + x2 - 7x4 = -8 b) Consider the linear system: { ax + bz = 2 ax + ay + 4z = 4 ay + 2z = b For which values of a does the linear system have i. a unique solution; ii. one-parameter infinitely many solutions; iii. two-parameter infinitely many solutions; iv. No solution. Q-4: [5+5 marks] a) Let S = {v1, v2, v3} be a linearly independent set of vectors in R^3. Determine whether T = {2v1 + v2 - 2v3, 4v1 + 3v2, 3v1 + 2v2 - v3} is linearly independent. b) Describe all vectors [a] that can be written as linear combinations of the [b] [c] vectors v1 = [4], v2 = [4], v3 = [0] [0] [2] [2] [-1] [2] [3]
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