# Sherman-Morrison formula

$LaTeX: \left [ A + a b^T \right ]^{-1} = A^{-1} - \frac{A^{-1} a b^T A^{-1}}{1 + b^T A^{-1} a},$
where $LaTeX: A$ is a nonsingular $LaTeX: n \times n$ matrix and $LaTeX: a$ and $LaTeX: b$ are n-vectors, provided the inverse (on the left) exists.