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  "body": "\n## Statement\n\nLet $A$ be an [[invertible matrix|0EB9N0S33C]] with inverse $B = A^{-1}$.\nThen for each vector $b$ of the right size, the [[system of linear equations|0JTRMBQ1SA]] $Ax = b$ is consistent.\nThe vector $x = Bb$ is a solution of it.\nSo the solution of $Ax = b$ is given by $x = A^{-1}b$.\n\n## Remarks\n\nThe check is direct: $A(Bb) = (AB)b = Ib = b$.\nThis is why an inverse, once known, solves every system with the same matrix $A$.\nUniqueness also holds, since $Ax = b$ gives $x = A^{-1}Ax = A^{-1}b$.\n",
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