Inverse of matrix sum, special case: $(A + x I)$

Question

Is there a simple way to do $(A + x I)^{-1}$ where $A$ is an invertible matrix, $I$ is unit matrix and $x$ is a scalar?

I see a lot of expressions for the general case $(A + B)^{-1}$, but nothing on the special case where $B = x I$.

I have to do $(A + x I)^{-1}$ for many $x$ values, so I want to know if some identity can be applied. Any help is highly appreciated. Thanks in advance.

Answer 1

For the similar expression $(I+xA)^{-1}$ we can use $$ (I+xA)^{-1}=I+xA+x^2A^2+x^3A^3+\ldots$$ provided $x^nA^n\to 0$ (for example if $A$ is nilpotent). Note that $A+xI = x\cdot(\frac1xA+I)$.