How to invert sum of matrices?

Question

Given are two matrices:

$\bf A, \bf B$

We know that matrices $\bf A \neq \bf B$ are invertable, symmetric, positive-definite and of full rank. Is it possible to give the formula for following sum of these matrices:

$[\bf A + \lambda\bf B]^{-1} = ?$

where $\lambda$ is a scalar such as $0 < \lambda < 1$.

Answer 1

Assuming that $A+\lambda B$ is also invertible, you can use the Binomial Inverse Theorem:

$[A+\lambda B]^{-1} = A^{-1}-\lambda A^{-1}(I + \lambda BA^{-1})^{-1}BA^{-1}$

Answer 2

Use Woodbury matrix identity:

$$ \left(A+UCV \right)^{-1} = A^{-1} - A^{-1}U \left(C^{-1}+VA^{-1}U \right)^{-1} VA^{-1}. $$