We establish necessary and sufficient conditions for invertibility of symmetric three-by-three block matrices having a double saddle-point structure \fb{that guarantee the unique solvability of double saddle-point systems}. We consider various scenarios, including the case where all diagonal blocks are allowed to be rank deficient. Under certain conditions related to the nullity of the blocks and intersections of their kernels, an explicit formula for the inverse is derived.

翻译：本文建立了对称三乘三块矩阵可逆性的充分必要条件，该矩阵具有双重鞍点结构，从而保证双重鞍点系统解的唯一性。我们考虑了多种情形，包括允许所有对角块均为秩亏缺的情况。在与各块零度及其核空间交集相关的特定条件下，我们推导出了逆矩阵的显式公式。