LU Decomposition

The LU decomposition of a square matrix $\mathbf{A}$ expresses it as $\mathbf{A}=\mathbf{LU}$, where $\mathbf{L}$ is lower triangular (with 1's on the diagonal, by convention) and $\mathbf{U}$ is upper triangular; in practice, a permutation matrix $\mathbf{P}$ is often needed, $\mathbf{PA}=\mathbf{LU}$, to allow row-swapping (partial pivoting) for numerical stability. Once a matrix is factored this way, solving $\mathbf{Ax}=\mathbf{b}$ reduces to two easy triangular solves (forward substitution for $\mathbf{L}$, then back substitution for $\mathbf{U}$) rather than one hard general solve, and the determinant follows immediately as the product of $\mathbf{U}$'s diagonal entries (times the permutation's sign), which is exactly how statistical software computes determinants and matrix inverses efficiently behind the scenes rather than through the far more expensive cofactor-expansion formula.

• Once a matrix is LU-factored, solving a linear system reduces to two simple triangular solves, forward substitution through $\mathbf{L}$ and back substitution through $\mathbf{U}$, each requiring only a straightforward top-to-bottom or bottom-to-top sweep, dramatically cheaper than solving the original system directly for a large matrix.
• Row pivoting (the permutation matrix $\mathbf{P}$) is included in practice not because it's mathematically required for every matrix, but because it protects against numerical instability, without it, dividing by a very small pivot entry during the factorization can amplify rounding error catastrophically, even for a matrix that is mathematically well-behaved.
• The determinant of $\mathbf{A}$ follows immediately once its LU decomposition is known, simply the product of $\mathbf{U}$'s diagonal entries (adjusted for the sign of any row permutations used), which is why LU decomposition, not the textbook cofactor-expansion formula, is the method actual statistical and numerical software uses to compute determinants for anything beyond a tiny matrix.