How does a change in the scalar $x$ affect the inverse $(A + xB)^{-1}$? $A$ and $B$ are both arbitrarily sized matrices of $N \times N$ with a known inverse. In this problem, $A^{-1}$ and $B^{-1}$ are known.
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I am trying to implement a control algorithm on an embedded system. The x represents the change in the time variable used to plan the trajectory. To save time, I am precomputing this matrix but still want to change my prediction horizon by varying my delta t. This x is typically less than 1 but the matrix B can become bigger than that
Assuming $A^{-1}$ exists, $(A+xB)^{-1} = A^{-1} (I + x B A^{-1})^{-1}$ exists whenever $-1/x$ is not an eigenvalue of $BA^{-1}$. Its matrix elements are rational functions of the complex variable $x$; at their poles, $-1/x$ is an eigenvalue of $BA^{-1}$.