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  "body": "\n## Statement\n\nLet $A$ be an $n \\times n$ matrix. The following four facts hold.\n\n1. The matrix $A$ is [[invertible|0EB9N0S33C]] if and only if $A$ is [[row equivalent|006BHFWMAP]] to the identity matrix $I$.\n2. If $A$ is invertible, then for each vector $\\mathbf b$ the equation $A\\mathbf x = \\mathbf b$ has the solution $\\mathbf x = A^{-1}\\mathbf b$.\n3. If $A$ is invertible, then row reducing the augmented matrix $[A \\mid I]$ gives its [[reduced row echelon form|0WF5AQYTW6]] $[I \\mid A^{-1}]$. This is a way to compute $A^{-1}$.\n4. If $A$ and $B$ are invertible $n \\times n$ matrices, then $AB$ is invertible and $(AB)^{-1} = B^{-1}A^{-1}$.\n\n## Remarks\n\nFact 2 follows by multiplying both sides of $A\\mathbf x = \\mathbf b$ on the left by $A^{-1}$.\n\nFact 4 is checked by the product $(AB)(B^{-1}A^{-1}) = I$.\n",
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