Linear Algebra quesion

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$A^{-1} - \lambda A = B^{-1} - \lambda B - \alpha v v^T$

$A, B \in S^n_+$; $v \in R^n$; $\lambda, \alpha \in R_+$. Can we solve A in term of other variables?

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Multiply by $A$.

You will get something like: $aA^2+bA+c=0$.