Y∼N₈(0,I₈). The quadratic forms YᵀΣ₁Y and YᵀΣ₂Y are independent central chi-square variables with 3 and 4 degrees of freedom. P:Σ₁ and Σ₂ are idempotent. Q:Σ₁Σ₂=0. Which hold?
Under the usual symmetric quadratic-form-matrix convention,a standard-normal quadratic is central χ² precisely when its matrix is an orthogonal projection of the corresponding rank. Thus both matrices are idempotent. Independence of symmetric Gaussian quadratic forms implies their product is zero. Hence the official answer is **C**. Symmetry is implicit:arbitrary skew-symmetric additions leave YᵀΣY unchanged,so without that convention neither printed matrix conclusion follows from the quadratic-form laws alone.
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