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Math Help - Gaussian elimination- Matrix row operations

  1. #1
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    Gaussian elimination- Matrix row operations

    Hi, one one of the elementary row operations, is the addition of a multiple of on equation to another equation. This operation will not change the value of a matrix, but I do not understand why. I would also like to know, if there is any graphical representation, to illustrate this concept.

    Thanks for your help”
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    Quote Originally Posted by unamofa View Post
    Hi, one one of the elementary row operations, is the addition of a multiple of on equation to another equation. This operation will not change the value of a matrix, but I do not understand why. I would also like to know, if there is any graphical representation, to illustrate this concept.

    Thanks for your help”

    What do you mean by "the value of the matrix"?? Did you mean "the value of the determinant of the (square, of course) matrix? If so this is due to the fact that determinant is a multilinear alternating function...

    Tonio
    Last edited by tonio; January 22nd 2010 at 07:11 AM. Reason: typo
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    Hi Tonio,

    I mean the value of the solution of the system.
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    Quote Originally Posted by unamofa View Post
    Hi Tonio,

    I mean the value of the solution of the system.
    It is clear from the way we reduce a system that if a certain set of numbers x_1,...,x_n satisfies the original system, then they also satisfy the reduced system. Turn the roles of original & reduced matrices around: If we start by the reduced system, we can obtain the original system by a combination of elementary row operations. Then clearly, any solution of the reduced system is also a solution of the original.

    Therefore, it's proved that elementary row operations, when applied to the matrix of a linear system, do NOT change the solution set of that system.
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  5. #5
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    Another way of looking at it: think of the system of equations the matrix represents. The three row- operations correspond to operations on the equations:
    "Swap two rows" is just "swap the positions of two equations". Obviously that will not change the solution set of the equations.

    "Multiply a row by a (non-zero) constant" is "multiply each term of the equation by the same number". Since you are multiplying both sides of the equation by the same thing, this does not change the solution set.

    "Add a multiple of one row to another" is a bit more complicated. It is a combination of "multiply one equation by a constant" and "add this new equation to another equation in the set". The first does not change the solution set because we are multiplying both sides of the equation by the same thing. The second does not change the solution set because both sides of the first equation are equal and so we are adding the same thing to both sides of the second equation.
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