1. The problem is to perform LU factorization of a given square matrix $A$. 2. LU factorization decomposes matrix $A$ into the product of a lower triangular matrix $L$ and an upper triangular matrix $U$, such that $$A = LU.$$ 3. To find $L$ and $U$, we use Gaussian elimination steps without row exchanges (assuming $A$ is nonsingular and can be factorized this way). 4. For example, if $$A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix},$$ we perform elimination to zero out elements below the main diagonal in $U$ and record multipliers in $L$. 5. The matrix $L$ has ones on the diagonal and the multipliers below the diagonal, while $U$ is the resulting upper triangular matrix after elimination. 6. This factorization is useful for solving linear systems $Ax=b$ efficiently by solving $Ly=b$ and then $Ux=y$. 7. If you provide a specific matrix, I can demonstrate the step-by-step LU factorization process.