Q:

Standard Matrix Co-ordinates Must Be Less Than 1?

(I can't post images until I have at least 10 reputation, so bear with me here).
There's a question in my textbook that goes:
http://dl.dropbox.com/u/23889576/Screenshots/StackExchange/1350.png
The answer that's shown states:
http://dl.dropbox.com/u/23889576/Screenshots/StackExchange/1351.png
Why does the -25/13 convert to -12/13 in the final answer? Is it true that a standard matrix cannot have co-ordinates greater than 1?

A:

It's a typo in the book. Notice that
$$1-5\frac{10}{26}=-\frac{12}{13}$$
The $-25/13$ is simply a mistake.
You can check this. Suppose the matrix actually should include $-25/13$, then it should fix the line $x_1-5x_2=0$. Well, consider a vector $[5,1]^{\top}$:
$$\left[\begin{array}{cc} 12/13 & 5/13 \\ 5/13 & -25/13\end{array}\right]\left[\begin{array}{c} 5 \\ 1\end{array}\right] = \left[\begin{array}{c} 5 \\ 0\end{array}\right]$$
Clearly $x_1=5$ and $x_2=0$ does not satisfy $x_1-5x_2=0$. The matrix does not fix the line.

