2-3 - Chapter 2.3: Gauss-Jordan Elimination Systems with an...

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Unformatted text preview: Chapter 2.3: Gauss-Jordan Elimination Systems with an Infinite Number of Solutions Today we will look at systems where there are an infinite num- ber of solutions, or no solution at all. Also, we will look at systems in which the number of equations is not equal to the number of variables. In all cases, the means of finding a solution are the same: 1. Write down the corresponding augmented matrix for the given system. 2. Row-reduce the matrix until it is in row-reduced form. 3. Determine what the solutions are. The only thing that changes is step three determining whether there is one solution, an infinite number of solutions, or no solution. Example: Solve the following system: x + 2 y = 3 4 x + 8 y = 20 First, we write down the corresponding augmented matrix: bracketleftbigg 1 2 3 4 8 20 bracketrightbigg Now, we row-reduce: bracketleftbigg 1 2 3 4 8 20 bracketrightbigg R 2- 4 R 1 bracketleftbigg 1 2 3 0 0 8 bracketrightbigg At this point, I claim that the matrix is in row-reduced form. Go back to the four rules for row-reduced form; you should confirm that this matrix satisfies all four conditions. Now, look at the last row of the matrix. The corresponding equation is 0 x + 0 y = 8, or 0 = 8. This is a contradiction. As before, when we end up with a contradiction, we conclude that the system has no solution. 1 If there is a row of zeros in the coefficient matrix, with a non-zero number in the augmented column, then the given system has no solution. Example: Solve the given system. x + y + 3 z = 2 2 x + 3 y + z = 9 3 x + 4 y + 4 z = 11 We begin, as always by writing down the corresponding aug- mented matrix, and row-reducing until the matrix is in row-reduced form. 1 1 3 2 2 3 1 9 3 4 4 11 R 2- 2 R 1 1 1 3 2 0 1- 5 5 3 4 4 11 R 3- 3 R 1 1 1 3 2 0 1- 5 5 0 1- 5 5 R 1- R 2 1 0 8- 3 0 1- 5 5 0 1- 5 5 R 3- R 2...
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This note was uploaded on 02/05/2011 for the course MATH 377 taught by Professor Stephenlang during the Spring '11 term at University of Victoria.

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2-3 - Chapter 2.3: Gauss-Jordan Elimination Systems with an...

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