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  "body": "\n## Statement\n\nLet $A$ be a square matrix.\nThen $A$ is [[invertible|0EB9N0S33C]] if and only if its [[reduced row echelon form|0WF5AQYTW6]] is the identity matrix $I$, that is, $A \\sim I$.\n\nMoreover, when $A$ is invertible, row reduction finds the inverse.\nForm the [[augmented matrix|0T6MWGZSXA]] $[A \\mid I]$ and apply [[Gauss–Jordan elimination|098D6FQN3N]].\nThe result is\n$$[A \\mid I] \\sim [I \\mid A^{-1}].$$\n\n## Remarks\n\nEach row operation is left multiplication by an invertible matrix.\nSo reducing $A$ to $I$ gives a product $E$ of such matrices with $EA = I$.\nApplying the same operations to $I$ gives $EI = E = A^{-1}$.\n",
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