In Lay is given the following problem:

Suppose the system below is consistent for all possible values offandg. What can you say about the coefficientscandd? Justify your answer.

$\displaystyle x_{1} + 3x_{2} = f$

$\displaystyle cx_{1} + dx_{2} = g$

In the answers is the following:

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

$\displaystyle \begin{bmatrix} 1 & 3 & f \\ 0 & d-3c & g-cf \end{bmatrix}$ shows thatd-3cmust be nonzero, sincefandgare arbitrary. Otherwise for some choices offandgthe second row could correspond to an equation of the form0=b, wherebis nonzero. Thus $\displaystyle d \neq 3c$.

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:

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

Row reduces to this:

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

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