LU Factorization

$\left(
\begin{array}{ccc}
1 & 2 & 3 \\
2 & 6 & 10 \\
3 & 10 & 12
\end{array}
\right) =
\left(
\begin{array}{ccc}
1 & 0 & 0 \\
2 & 1 & 0 \\
3 & 2 & 1
\end{array}
\right) \cdot
\left(
\begin{array}{ccc}
1 & 2 & 3 \\
0 & 2 & 4 \\
0 & 0 & -5
\end{array}
\right)
$
or
$A=L.U$ where $A$ is a non-singular matrix and $L,U$ are respectively lower- and upper-triangular matrices.

Every non-singular matrix can be factorized in the product of a lower- and upper-triangular matrix. ( The factorization itself is trivial. )