On a determinant involving linear combinations of Legendre symbols

In this paper, we prove a conjecture of the second author by evaluating the determinant $$\det {[x+\left(\frac{i-j}{p}\right)+\left(\frac{i}{p}\right)y+\left(\frac{j}{p}\right)z+\left(\frac{ij}{p}\right)w]}_{0\le i,j\le (p-3)\mathrm{/}2}$$ for any odd prime $p$, where $(\frac{\cdot }{p})$ denotes the Legendre symbol. In particular, the determinant is equal to $x$ when $p\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)$.