LU decomposition forms the backbone of other algorithms in linear algebra, such as the solution of linear equations given by $$ \begin{align} a_{11}x_1 +a_{12}x_2 +a_{13}x_3 + a_{14}x_4=&w_1 \nonumber \\ a_{21}x_1 + a_{22}x_2 + a_{23}x_3 + a_{24}x_4=&w_2 \nonumber \\ a_{31}x_1 + a_{32}x_2 + a_{33}x_3 + a_{34}x_4=&w_3 \nonumber \\ a_{41}x_1 + a_{42}x_2 + a_{43}x_3 + a_{44}x_4=&w_4. \nonumber \end{align} $$ The above set of equations is conveniently solved by using LU decomposition as an intermediate step.
The matrix \( \mathbf{A}\in \mathbb{R}^{n\times n} \) has an LU factorization if the determinant is different from zero. If the LU factorization exists and \( \mathbf{A} \) is non-singular, then the LU factorization is unique and the determinant is given by $$ det\{\mathbf{A}\}=det\{\mathbf{LU}\}= det\{\mathbf{L}\}det\{\mathbf{U}\}=u_{11}u_{22}\dots u_{nn}. $$