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  "body": "\n## Summary\n\nGaussian elimination is an algorithm that changes a matrix into a [[row echelon form|0VYCDGMJJJ]] by [[elementary row operations|04XWH28452]].\nIt works column by column, from left to right.\nIn each column, it chooses a nonzero entry as [[pivot|0J0BBCMJ5W]] and moves its row up.\nThen it adds multiples of that row to the rows below, to make their entries in that column zero.\nApplied to the [[augmented matrix|0T6MWGZSXA]] of a [[system of linear equations|0GT4WWGWH2]], it gives an equivalent system that [[back substitution|0ATFVE08MW]] solves.\nFor an $n \\times n$ matrix, it uses on the order of $n^3$ arithmetic operations.\n\n## Remarks\n\nThe algorithm also computes the [[rank|0ZC9YTXZRY]] of a matrix and, with a record of the row operations, its [[determinant|00KDZRW4C0]].\nIt also tests if a system is [[consistent|0X6V9B1V10]]: the system is inconsistent exactly when a row $(0, \\dots, 0 \\mid c)$ with $c \\ne 0$ appears.\nThe method was known in China in the book *The Nine Chapters on the Mathematical Art*, long before Gauss.\n\n## Explanations\n\n### L1 human\n\nGaussian elimination is the method of solving equations by removing one unknown at a time.\nWe use the first equation to remove $x_1$ from all the other equations, then the second to remove $x_2$, and so on.\nAt the end, the last equation has the fewest unknowns, and we solve upwards.\n",
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