# Invertibility and solution of equation with compound circular matrix

Given the matix C, which can be represented as: C=AP + BQ (C=(A|B)), where A and B are circulant matrices; P and Q are projectors (P projects first half of matrix vectors, Q projects second half). C consists of two circular matricex. A is invertible matrix. C also can be represented as: $$C=A(P+A^{-1}BQ)$$. All matrix is invertible, if $$(P+A^{-1}BQ)$$ is invertible. So, we have scheme of matrix: \begin{bmatrix} I & PA^{-1}BQ \\ O & QA^{-1}BQ \\ \end{bmatrix} Where I - identity matrix, O is zero matrix. How can I obtain the conditions of reversibility and solve equation Cx=b ?