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Math Help - Info about coefficients of augmented matrix

  1. #1
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    Info about coefficients of augmented matrix

    In Lay is given the following problem:

    Suppose the system below is consistent for all possible values of f and g. What can you say about the coefficients c and d? Justify your answer.

    x_{1} + 3x_{2} = f

    cx_{1} + dx_{2} = g

    In the answers is the following:

    The row reduction of \begin{bmatrix} 1 & 3 & f \\ c & d & g \end{bmatrix} to

    \begin{bmatrix} 1 & 3 & f \\ 0 & d-3c & g-cf \end{bmatrix} shows that d-3c must be nonzero, since f and g are arbitrary. Otherwise for some choices of f and g the second row could correspond to an equation of the form 0=b, where b is nonzero. Thus d \neq 3c.

    This is just fine, what I don't understand however is the missing working between the initial matrix and the row reduced version. I can't even figure out which row operations to begin with.

    Can someone please explain how this matix:

    \begin{bmatrix} 1 & 3 & f \\ c & d & g \end{bmatrix}

    Row reduces to this:

    \begin{bmatrix} 1 & 3 & f \\ 0 & d-3c & g-cf \end{bmatrix}

    ty
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  2. #2
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    Re: Info about coefficients of augmented matrix

    The second row is replaced by the sum of the second row and the product of the first row times -c.
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  3. #3
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    Re: Info about coefficients of augmented matrix

    That makes sense in such a simple way that it leaves me feeling very stupid.
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