Properties of Determinants

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

 $\left|\begin{array}{ccc}x+a& x& x\\ x& x+a& x\\ x& x& x+a\end{array}\right|=0$

The value of the following determinant would be:

 $\left|\begin{array}{ccc}a-b& b-c& c-a\\ b-c& c-a& a-b\\ c-a& a-b& b-c\end{array}\right|$

If  $\alpha , \beta , \gamma$  are in A.P then, what would be the value of determinant  $\left|\begin{array}{ccc}x-3& x-4& x-\alpha \\ x-2& x-3& x-\beta \\ x-1& x-2& x-\gamma \end{array}\right|$

If  $\alpha ,\hspace{0.17em}\beta ,\hspace{0.17em}\gamma$  are in A.P then, what would be the value of determinant  $\left|\begin{array}{ccc}x-3& x-4& x-\alpha \\ x-2& x-3& x-\beta \\ x-1& x-2& x-\gamma \end{array}\right|$

Using properties of determinants, the solution for x:  $\left|\begin{array}{l}a+xa-xa-x\\ a-xa+xa-x\\ a-xa-xa+x\end{array}\right|=0,$ would be :

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

Which of the following operation can prove,  $\left|\begin{array}{ccc}\sin \alpha & \cos \alpha & \cos \left(\alpha +\delta \right)\\ \sin \beta & \cos \beta & \cos \left(\beta +\delta \right)\\ \sin \gamma & \cos \gamma & \cos \left(\gamma +\delta \right)\end{array}\right|=0$

Using properties of determinants, the value of   $\left|\begin{array}{ccc}3a& -a+b& -a+c\\ a-b& 3b& c-b\\ a-c& b-c& 3c\end{array}\right|$ would be

       Let  $\omega =-\frac{1}{2}+i\frac{\sqrt{3}}{2}.$  then the value of  $\left|\begin{array}{l}111\\ 1-1-{\omega }^2{\omega }^2\\ 1{\omega }^2{\omega }^4\end{array}\right|$ is equal to –