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Linear_Systems_2

# Linear_Systems_2 - Linear Systems(1 Use the Gauss-Jordan...

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Linear Systems (1) Use the Gauss-Jordan method to solve: x 1 - x 2 + x 3 + 2 x 4 = 7 2 x 1 - x 2 + 3 x 4 = 12 - x 1 + x 2 + x 3 - 2 x 4 = - 7 (2) Given the linear system - x 1 + x 2 - x 3 - 2 x 4 = 0 5 x 1 - 3 x 2 + x 3 + 4 x 4 = 2 8 x 1 + 3 x 2 - 15 x 3 - 15 x 4 = 8 (a) Find the general solution by using Gauss-Jordan elimination. (b) Find the particular solution for which x 1 = 7. (3) 2 x 1 - 3 x 2 + x 3 = 7 x 1 - 2 x 2 - x 3 + x 4 = 2 3 x 1 - 5 x 2 - x 3 + 2 x 4 = 9 (a) Find all solutions of the given system. (b) If the system has more than one solution, find a particular solution. (c) Can you use Cramer’s rule to solve this system? Explain. (4) Solve the following linear system. If there is more than one solution give a particular solution. 2 x 1 - 3 x 2 + x 3 = 6 2 x 1 - 2 x 2 - x 3 + 3 x 4 = 1 3 x 1 - 5 x 2 - x 3 + 2 x 4 = 8 (5) Find the general solution to the homogeneous linear system: x 1 + 3 x 2 - 4 x 3 + 2 x 4 = 0 2 x 1 + 7 x 2 - 6 x 3 = 0 - 3 x 1 - 8 x 2 + 14 x 3 - 10 x 4 = 0 (6) Solve by using the Gauss-Jordan elimination x 1 + 2 x 2 - 2 x 3 + x 4 = 2 2 x 1 - x 2 + x 3 + 7 x 4 = - 1 - 3 x 1 + x 3 + x 4 = 1 1

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2 (7) Solve the linear system using Gauss-Jordan method: x 1 + 2 x 2 + 3 x 3 - x 4 = 3 2 x 1 + 3 x 2 + x 3 = 9 3 x 1 + 5 x 2 + 2 x 3 - 3 x 4 = 10 (8) Find conditions on k and l so that the system x + y + 2 z = 4 2 x + 3 y - z = 4 3 x + 4 y - kz = l has (a) unique solution (b) infinitely many solutions
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