Inverse of sum of scalar times matrix plus identity matrix I'm trying to figure out a way of seperating out my $n\times n$-dimensional matrix $A$ (to use in my Finite Difference scheme) from the following equation:

$(cA + I)^{-1}$,

where $c$ is a scalar and $I$ a $n\times n$ identity matrix.

Does anyone have a (reference to a) clue how to proceed and, if possible, under which assumptions a solution can be obtained?
I'm a bit confused; if you have a matrix $M$ and you want to write it in the form
$$M = cA + I$$
you simply directly solve for $A$:
$$A = (M-I)/c$$
where if you're not given the value of $c$, you can pick it arbitrarily ($c=1$, say).



If you're trying to compute the matrix inverse of $cA+I$, that is a different problem (and one without a closed-form solution, though you can use some common approximations, e.g. $$(I+cA)^{-1} \approx I - cA,$$ if $cA$ is small).
I'm a bit confused; if you have a matrix $M$ and you want to write it in the form
$$M = cA + I$$
you simply directly solve for $A$:
$$A = (M-I)/c$$
where if you're not given the value of $c$, you can pick it arbitrarily ($c=1$, say).



If you're trying to compute the matrix inverse of $cA+I$, that is a different problem (and one without a closed-form solution, though you can use some common approximations, e.g. $$(I+cA)^{-1} \approx I - cA,$$ if $cA$ is small).