We establish necessary and sufficient conditions for invertiblility of symmetric three-by-three block matrices having a double saddle-point structure 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 ranks of the blocks and intersections of their kernels, an explicit formula for the inverse is derived.

翻译：我们建立了具有双重鞍点结构的对称三乘三分块矩阵可逆的充要条件，这些条件保证了双重鞍点系统的唯一可解性。我们考虑了多种情形，包括所有对角块允许秩亏缺的情况。在与各分块秩及其核的交集相关的特定条件下，推导出了逆矩阵的显式公式。