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In this paper, we confirm some conjectures posed by Sun and investigate some related topics.\n  For instance, given any integers $c,d$ with $d\\ne0$ and $c^2-4d\\ne0$, we show that there are infinitely many odd primes $p$ such that $$\\det\\bigg[\\left(\\frac{i^2+cij+dj^2}{p}\\right)\\bigg]_{0\\le i,j\\le p-1}=0,$$ where $(\\frac{\\cdot}{p})$ is the Legendre symbol. 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