Is the Determinant of a Symmetric Matrix with Zero Diagonal Elements Non-Zero?

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The discussion focuses on proving that the determinant of a symmetric matrix with zero diagonal elements and all other positive elements is non-zero. Participants explore the implications of the matrix's structure, emphasizing that the positive off-diagonal elements contribute to a non-zero determinant. The conversation highlights the importance of matrix properties in linear algebra, particularly regarding symmetry and positivity. Clarifications are made regarding the term "different," indicating that the off-diagonal elements must be distinct. Overall, the consensus is that such a matrix indeed has a non-zero determinant.
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How to prove that the determinant of a symmetric matrix with the main diagonal elements zero and all other elements positive is not zero and different ?
 
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different from what?
 
all different element
 

Thread 'Confusion about the Moyal-Weyl twist'

I am studying the mathematical formalism behind non-commutative geometry approach to quantum gravity. I was reading about Hopf algebras and their Drinfeld twist with a specific example of the Moyal-Weyl twist defined as F=exp(-iλ/2θ^(μν)∂_μ⊗∂_ν) where λ is a constant parametar and θ antisymmetric constant tensor. {∂_μ} is the basis of the tangent vector space over the underlying spacetime Now, from my understanding the enveloping algebra which appears in the definition of the Hopf algebra...
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