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  "body": "\n## Summary\n\nRow reduction computes the inverse of a square matrix, or shows that no inverse exists.\nLet $A$ be an $n \\times n$ matrix over a field $F$, and form the $n \\times 2n$ [[augmented matrix|0T6MWGZSXA]] $[A \\mid I_n]$.\nApply [[Gauss–Jordan elimination|098D6FQN3N]] to it.\nIf the left block becomes $I_n$, then the result is $[I_n \\mid A^{-1}]$, and the right block is the inverse.\nIf the left block gets a zero row, then $A$ is not [[invertible|0EB9N0S33C]].\n\n## Remarks\n\nThe method works because each row operation is left multiplication by an [[elementary matrix|01REXW3WWE]].\nIf $E_k \\cdots E_1 A = I_n$, then $E_k \\cdots E_1 = A^{-1}$, and the same product changes $I_n$ into $A^{-1}$.\nFor $A = \\begin{pmatrix} 2 \u0026 1 \\\\ 1 \u0026 1 \\end{pmatrix}$, the method gives $A^{-1} = \\begin{pmatrix} 1 \u0026 -1 \\\\ -1 \u0026 2 \\end{pmatrix}$.\n\n## Explanations\n\n### L1 human\n\nWe write the identity matrix next to $A$ and row reduce the whole wide matrix.\nWhen the left half becomes the identity, the right half has become $A^{-1}$.\nThe same row operations that turn $A$ into $I$ turn $I$ into $A^{-1}$.\n",
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