I have the following matrix I am trying to decompose into it's respective $L$ and $U$ parts for $A = LU$.
So I have $$\begin{bmatrix} 1 && 4 && 3 \\ 0 && -10 && -5 \\ 0 && -8 && -4 \end{bmatrix}$$
$$ \left[ \begin{array}{cccccc|cccccc} 1 && 4 && 3 && && 1 && 0 && 1\\ 0 && -10 && -5 && && 0 && 1 && 0\\ 0 && -8 && -4 && && 0 && 0 && 1 \end{array} \right] $$
I notice that column $1$ is fine, and I need to make $R_3 C_2 = 0$ to achieve the upper triangle matrix on the left and the lower triangle matrix on the right.
So I do: $R_3 \longrightarrow -\frac{5}{4} R_3 + R_2$
$$ \left[ \begin{array}{cccccc|cccccc} 1 && 4 && 3 && && 1 && 0 && 1\\ 0 && -10 && -5 && && 0 && 1 && 0\\ 0 && 0 && 0 && && 0 && 1 && -5/4 \end{array} \right] $$
However, the answer key is telling me the answer is:
$$ \left[ \begin{array}{cccccc|cccccc} 1 && 4 && 3 && && 1 && 0 && 1\\ 0 && -10 && -5 && && 0 && 1 && 0\\ 0 && -8 && -4 && && 0 && 4/5 && 1 \end{array} \right] $$