Using properties of determinants, prove that

$\left|\begin{array}{ccc}1& \alpha & {\alpha }^2+\beta \gamma \\ 1& \beta & {\beta }^2+\gamma \alpha \\ 1& \gamma & {\gamma }^2+\alpha \beta \end{array}\right|=2(\alpha -\beta )(\beta -\gamma )(\gamma -\alpha )$.

Using properties of determinant, prove that

$\left|\begin{array}{ccc}{a}^2+1& ab& ac\\ ab& b^2+1& bc\\ ac& bc& c^2+1\end{array}\right|=\left|\begin{array}{ccc}{a}^2+1& b^2& c^2\\ a^2& b^2+1& c^2\\ a^2& b^2& c^2+1\end{array}\right|=1+a^2+b^2+c^2$.

Using the properties of determinant, prove that

$\left|\begin{array}{ccc}a& b& c\\ a^2& b^2& c^2\\ bc& ca& ab\end{array}\right|=\left|\begin{array}{ccc}1& a^2& a^3\\ 1& b^2& b^3\\ 1& c^2& c^3\end{array}\right|=\left(a-b\right)\left(b-c\right)\left(c-a\right)\left(ab+bc+ca\right)$.

Using properties of determinants, prove that

$\left|\begin{array}{ccc}{a}^2& a^2-(b-c{)}^2& bc\\ b^2& b^2-(c-a{)}^2& ca\\ c^2& c^2-(a-b{)}^2& ab\end{array}\right|=\left(a-b\right)\left(b-c\right)\left(c-a\right)\left(a+b+c\right)\left(a^2+b^2+c^2\right)$.

Using properties of determinants, prove that

$\left|\begin{array}{ccc}(b+c{)}^2& a^2& bc\\ (c+a{)}^2& b^2& ca\\ (a+b{)}^2& c^2& ab\end{array}\right|=(a^2+b^2+c^2)(a+b+c)(b-c)(c-a)(a-b)$.

Using properties of determinants, prove that

$\left|\begin{array}{ccc}a+b+c& -c& -b\\ -c& a+b+c& -a\\ -b& -a& a+b+c\end{array}\right|=2(a+b)(b+c)(c+a)$.

Using properties of determinant, prove that

$\left|\begin{array}{ccc}y+z& x+y& x\\ z+x& y+z& y\\ x+y& z+x& z\end{array}\right|=x^3+y^3+z^3-3xyz$.

Using properties of determinants, prove that

$\left|\begin{array}{ccc}a& b-c& c-b\\ a-c& b& c-a\\ a-b& b-a& c\end{array}\right|=(a+b-c)(b+c-a)(c+a-b)$.

Using properties of determinants, prove that

$\left|\begin{array}{ccc}1& bc+ad& b^2{c}^2+a^2{d}^2\\ 1& ca+bd& c^2{a}^2+b^2{d}^2\\ 1& ab+cd& a^2{b}^2+c^2{d}^2\end{array}\right|=\left(a-b\right)\left(a-c\right)\left(a-d\right)\left(b-c\right)\left(b-d\right)\left(c-d\right)$.