# Info about coefficients of augmented matrix

• Dec 5th 2011, 11:40 AM
ecolitan
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
• Dec 6th 2011, 12:46 AM
emakarov
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.
• Dec 6th 2011, 01:55 AM
ecolitan
Re: Info about coefficients of augmented matrix
That makes sense in such a simple way that it leaves me feeling very stupid.