I am working on a simultaneous linear equation problem using LU decomposition and I'm unsure if this is the correct approach/answer to solve a system of simultaneous equations using LU decomposition. I'm looking for a way to check my resulting calculations and to understand if I have done the calculations correctly. This is my first time doing LU decomposition or doing matrix calculations at this level.
$5x1 + 6x2 + 2.3x3 + 6x4 = 4$
$9x1 + 2x2 + 3.5x3 + 7x4 = 5$
$3.5x1 + 6x2 + 2x3 + 3x4 = 6.7$
$1.5x1 + 2x2 + 1.5x3 + 6x4 = 7.8$
\begin{bmatrix}5&6&2.3&6\\9&2&3.5&7\\3.5&6&2&3\\1.5&2&1.5&6\end{bmatrix}
I multiply the top, third and fourth row by 10 and the second by 2 to make it easier to work with.
\begin{bmatrix}50&60&23&60\\18&4&7&14\\35&60&20&30\\15&20&15&60\end{bmatrix}
I calculated the L matrix as:
\begin{bmatrix}1&0&0&0\\9/25&1&0&0\\7/10&45/44&1&0\\3/10&5/44&1.584572&1\end{bmatrix}
and U matrix as: \begin{bmatrix}50&60&23&60\\0&88/25&-1.28&-38/5\\0&0&5.2&-93/22\\0&0&0&12.6\end{bmatrix}
I have y as the following: \begin{bmatrix}4\\5\\6.7\\7.8\end{bmatrix}
and solved the system with the following values for x \begin{bmatrix}4\\3.56\\0.2590\\5.734906\end{bmatrix}