Explain in your own words how to solve linear systems by using Gauss-Jordan Elimination. Be specific and include how to write the solution set of the system from the resulting matrix. The central theme of Gauss Jordan Elimination method of solving a system of linear equations can be summarized as below: (a) Write the augmented matrix [A|b] for the given system of linear equations. (b) Use elementary row operations on the augmented matrix [A|b] to transform A into diagonal form (that is, we want to have zeroes both above and below the diagonal). If a zero is located on the diagonal, switch the rows until a nonzero element comes in that place. If it is not possible to do this, then either the system has infinitely many solutions or no solution. (c) Divide the diagonal element and the RHS element in each row by the diagonal element in that row, and make each diagonal element equal to one. (d) Read the solution from the last matrix. An example will make the whole process very clear: Solve: 2y + z = 4 x + y + 2z = 6 2x + y + z = 7 Solution: (b) Interchange R1 and R2 R3 - 2R1 R1 - (1/2)R2 R3 + (1/2)R2 R1 + (3/5)R3 R2 + (2/5)R3 (c) (1/2)R2; (-2/5)R3 (d) The solution is: x = 11/5, y = 7/5, z = 6/5.
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