LU Factorization - Linear Algebra LU-factorization

My solution:

Am I on the right path? Or am I completely off?
 A: In copying the matrix $A$, you seem to have changed the sign of the last entry of the first row!
In your second step, $R_3 \to R_3 + 4 R_1$, you have neglected to replace the first entry of $R_3$ by 4 (which is 4 times the first entry of $R_1$).  Since you don't want to get nonzero entries in the early columns, you shouldn't use $R_1$ for this. Instead, try $R_3 \to R_3 - 2 R_2$.
It is easy to check if you have a correct decomposition: Just multiply your $L$ and $U$ matrices and see if you get $A$ back!
In this case, you get
$$
\begin{bmatrix}
1 & -1 & 3 & -2 \\
0 & 2 & -1 & 2 \\
3 & -11 & 14 & -15
\end{bmatrix}
$$
which is not the desired outcome.
A: You seem to be using a modification of the Doolitle algorithm, but some steps or data are missing, because the last matrix multiplication does not give the correct result: e.g. in the right bottom corner your matrices give $3*(-2) + (-4)*2 + 1*(-1) = -15$, and not $1$.
While was checking the calculations, a correct answer was given by Kundor. Just in case you are interested, here is what I get using SMath Studio:
 
