= Solution
The given real symmetric <matrix> is <Hermitian> and has <trace> one. A <spectral decomposition> supplies its positivity without relying on diagonal entries alone:
$$
\rho=\frac34|v_1\rangle\langle v_1|
+\frac14|v_2\rangle\langle v_2|,\qquad
v_1=\frac{(2,1,1)^{\mathsf T}}{\sqrt6},\quad
v_2=\frac{(0,1,-1)^{\mathsf T}}{\sqrt2}.
$$
The remaining normalized <eigenvector> is $(-1,1,1)^{\mathsf T}/\sqrt3$ with <eigenvalue> zero. Consequently the spectrum is \b[$\{3/4,1/4,0\}$], every <eigenvalue> is nonnegative, and the <matrix> is permissible. Its <purity> is $(3/4)^2+(1/4)^2=\boxed{5/8}$, so it is \b[mixed].
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