# How does adding extra row and column of ones affect a matrix's inverse?

I'm working on a homework problem, and I'm stuck. I guess my linear algebra still needs some work...

I've arrived at

$\mathbf{D}= \left[ \begin{matrix} \mathbf{C} & \mathbf{1}^T \\ \mathbf{1} & 0 \end{matrix} \right]$

where $\mathbf{C}$ is a $n$ by $n$ matrix and $\mathbf{1}$ is a $n$ by $1$ vector of all ones. I need to find $\mathbf{D}^{-1}$. Can I express it in terms of $\mathbf{C}^{-1}$? Can I proceed at all? Does it help if $\mathbf{C}$ is symmetric?

Thanks

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Have you tried any examples? Make up some $2\times2$ matrix $C$, and see how/whether $D^{-1}$ relates to $C^{-1}$? –  Gerry Myerson Sep 18 '12 at 6:11
I have tried that. It's difficult to see any relation... but I guess I don't know what to look for. –  Paul Accisano Sep 18 '12 at 6:13
Look up the "Schur complement", or see my comment on math.stackexchange.com/questions/182309/… –  copper.hat Sep 18 '12 at 6:36
Computing the Schur complement requires that $D$ (ie, the component in the lower right) be invertable, which it isn't... EDIT: Or at least that's what my cursory reading of the wiki article says. Your linked comment is much more helpful. Thanks! –  Paul Accisano Sep 18 '12 at 6:40
How is it that you're asking how to find $\mathbf D^{-1}$ if you believe that $\mathbf D$ isn't invertible? –  joriki Sep 18 '12 at 6:42