In mathematics, which condition guarantees that a real symmetric matrix is positive definite?

Answer and explanation

Correct answer: D. All its eigenvalues are strictly positive

  1. Step 1: By the spectral theorem, a real symmetric matrix has real eigenvalues and is positive definite if and only if all its eigenvalues are strictly positive (or equivalently all leading principal minors are positive). Answer: All its eigenvalues are strictly positive. Why not the others: a zero determinant implies singularity, and negative diagonal entries immediately violate positive definiteness.

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