Use properties of determinants to solve for $x$:

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

If $s=a+b+c$, then prove that

$\left|\begin{array}{ccc}s+c& a& b\\ c& s+a& b\\ c& a& s+b\end{array}\right|=2s^3$.

Using the properties of determinant, prove that 

$\left|\begin{array}{ccc}x-3& x-4& x-\alpha \\ x-2& x-1& x-\beta \\ x-1& x-2& x-\gamma \end{array}\right|=0$, where $\alpha , \beta and \gamma$ are in $\mathrm{A}.\mathrm{P}.$

Using properties of determinant, prove that

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

Using properties of determinant, prove that

$\left|\begin{array}{ccc}1+a& b& c\\ a& 1+b& c\\ a& b& 1+c\end{array}\right|=1+a+b+c$.

Using properties of determinant, prove that

$\left|\begin{array}{ccc}a+x& y& z\\ x& a+y& z\\ x& y& a+z\end{array}\right|=a^2(a+x+y+z)$.

Using properties of determinants, prove that

$\left|\begin{array}{ccc}y+k& y& y\\ y& y+k& y\\ y& y& y+k\end{array}\right|=k^2(3y+k)$.

Using properties of determinants, prove that

$\left|\begin{array}{ccc}x+4& 2x& 2x\\ 2x& x+4& 2x\\ 2x& 2x& x+4\end{array}\right|=(5x+4)(4-x{)}^2$.

Using properties of determinant, prove that

$\left|\begin{array}{ccc}a+b+nc& na-a& nb-b\\ nc-c& b+c+na& nb-b\\ nc-c& na-a& c+a+nb\end{array}\right|=n(a+b+c{)}^3$.