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

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

Without expanding any of the determinants given below, prove that

$\left|\begin{array}{ccc}{a}^2& b^2& c^2\\ (a+1{)}^2& (b+1{)}^2& (c+1{)}^2\\ (a-1{)}^2& (b-1{)}^2& (c-1{)}^2\end{array}\right|=4\left|\begin{array}{ccc}{a}^2& b^2& c^2\\ a& b& c\\ 1& 1& 1\end{array}\right|$.

Using the properties of determinants, solve the equation for $x$:

$\left|\begin{array}{ccc}3x-8& 3& 3\\ 3& 3x-8& 3\\ 3& 3& 3x-8\end{array}\right|=0$.

Using the properties of determinants, solve the equation for $x$:

$\left|\begin{array}{ccc}x+1& 3& 5\\ 2& x+2& 5\\ 2& 3& x+4\end{array}\right|=0$.

Using the properties of determinants, solve the equation for $x$:

$\left|\begin{array}{ccc}x& -6& -1\\ 2& -3x& x-3\\ -3& 2x& x+2\end{array}\right|=0$.

Using the properties of determinants, solve the equation for $x$:

$\left|\begin{array}{ccc}15-2x& 11-3x& 7-x\\ 11& 17& 14\\ 10& 16& 13\end{array}\right|=0$.

If $a,b,c$ are all different and $\left|\begin{array}{ccc}a& a^3& a^4-1\\ b& b^3& b^4-1\\ c& c^3& c^4-1\end{array}\right|=o$, then prove that

$abc(ab+bc+ca)=a+b+c$.