# Inverse of sum of 3 matrices

I need a way to compute the inverse of the sum of three matrices:

$(A + BB^T + \beta I)^{-1}$ where $I$ is identity and $\beta$ is a constant.

I am not very familiar with linear algebra, but a quick google search didn't help.

• Are these specific matrices, or just some general matrices? – Mike Pierce Jul 10 '15 at 2:20
• If you tell us why you want to find that, we could probably better help. – Spenser Jul 10 '15 at 2:24
• I don't think there is any reason to suppose that the inverse exists... – Alfred Yerger Jul 10 '15 at 2:30
• Let $C=BB^T+\beta I$ (which is symmetric; not sure if that helps). By [this][1] we have that \begin{align*} (A+C)^{-1}&= A^{-1}-(I+A^{-1}C)^{-1}A^{-1}CA^{-1} \end{align*} as long as $A$ is invertible. Not sure if that helps. [1]: math.stackexchange.com/questions/17776/… – Rocket Man Jul 10 '15 at 2:39
• I am doing a recursive least squares fit by hand, but i made some alterations to the original derivation, arrived at this step and couldnt continue. Let's assume the inverse exist. It should (i think). Idk if that helps... I might beable to just make python do this for me without using a closed form. But, i dont think so. – Xavier Hubbard Anderson Jul 10 '15 at 4:22