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  "body": "\n## Summary\n\nAn *LU decomposition* of a square matrix $A$ is a factorization $A = LU$.\nHere $L$ is a [[lower triangular matrix|00J3KZP8Y0]] and $U$ is an [[upper triangular matrix|0N4XY7Z3NN]].\n[[Gaussian elimination|0JVJ2WCXZH]] without row exchanges produces such a factorization: $U$ is the row echelon form, and $L$ stores the multipliers of the row operations.\nWith row exchanges, one gets $PA = LU$ for a [[permutation matrix|08SP3DF8NB]] $P$, and each square matrix over a field has such a factorization.\n\n## Remarks\n\nAn invertible matrix has an LU decomposition with $L$ unit lower triangular if and only if all its leading principal minors are nonzero.\nThis decomposition is then unique.\nThe matrix $\\begin{pmatrix} 0 \u0026 1 \\\\ 1 \u0026 0 \\end{pmatrix}$ has no LU decomposition, but it has one after a row exchange.\nTo solve $Ax = b$, solve $Ly = b$ by forward substitution and $Ux = y$ by [[back substitution|0ATFVE08MW]].\nThe decomposition also gives $\\det A = \\det L \\det U$.\n\n## Explanations\n\n### L1 human\n\nAn LU decomposition writes a matrix as a product of two triangular matrices.\nTriangular systems are easy to solve, so we solve $Ax = b$ in two easy steps.\nIt is useful when we solve many systems with the same $A$ and different right sides.\n",
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