Applications of saddle-point determinants

For a given square matrix $A ∈ M_n({ℝ})$ and the vector $e ∈ (ℝ)^{n}$ of ones denote by (A,e) the matrix
⎡ A e ⎤
⎣ $e^{T}$ 0 ⎦
This is often called the saddle point matrix and it plays a significant role in several branches of mathematics. Here we show some applications of it in: game theory and analysis. An application of specific saddle point matrices that are hollow, symmetric, and nonnegative is likewise shown in geometry as a generalization of Heron's formula to give the volume of a general simplex, as well as a conditions for its existence.