If $M$ is a $3 * 3$ matrix and $v$ is $3*1$ vector.
How can we prove $M^{-1}vv' = \frac{(M^{-1}v)(v'M^{-1})}{1 + v'M^{-1}v} (M+vv')$ ?
2026-03-31 03:33:50.1774928030
How can we prove LHS equals RHS of this matrix equation?
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I'll show you one step to begin with.
Pre-multiply by $M$ to obtain $$ vv'=\frac{vv'M^{-1}}{1+v'M^{-1}v}(M+vv')\\ vv'=\frac{vv'+vv'M^{-1}vv'}{1+v'M^{-1}v}. $$ Can you see how to go on from here? A hint is to get rid of the denominator...