Determinants Te...

If $$1, \omega, \omega^{2}$$ are the roots of unity then $$ \triangle =\left| \begin{matrix} 1 & { \omega  }^{ n } & { \omega  }^{ 2n } \\ { \omega  }^{ n } & { \omega  }^{ 2n } & 1 \\ { \omega  }^{ 2n } & 1 & { \omega  }^{ n } \end{matrix} \right|$$ is equal to-

If a matrix $$\begin{bmatrix} { \left( x-a \right)  }^{ 2 } & { \left( x-b \right)  }^{ 2 } & { \left( x-c \right)  }^{ 2 } \\ { \left( y-a \right)  }^{ 2 } & { \left( y-b \right)  }^{ 2 } & { \left( y-c \right)  }^{ 2 } \\ { \left( z-a \right)  }^{ 2 } & { \left( z-b \right)  }^{ 2 } & { \left( z-c \right)  }^{ 2 } \end{bmatrix}$$ is a zero matrix, then $$a,b,c,x,y,z$$ are connected by:

The determinant $$\Delta=\left| \begin{matrix} { a }^{ 2 }\left( 1+x \right)  & ab & ac \\ ab & { b }^{ 2 }\left( 1+x \right)  & bc \\ ac & bc & { c }^{ 2 }\left( 1+x \right)  \end{matrix} \right| $$ is divisible by  

The determinant $$\left| \begin{matrix} a & b & a\alpha +b \\ b & c & b\alpha +c \\ a\alpha +b & b\alpha +c & 0 \end{matrix} \right|$$ is equal to zero, if

If the points $$A(x, 2), B(-3, -4)$$ and $$C(7, -5)$$ are collinear, then the value of $$x$$ is :

Which of the following is/are true ? 
 (i)  Adjoint of a symmetric matrix is symmetric 
(ii)  Adjoint of a unit matrix is a unit matrix
(iii) A(adj A)=(adj A) A= [A]f and 
(iv) Adjoint of a diagonal matrix is a diagonal matrix  

If $$A=\begin{bmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \end{bmatrix}$$, then the value of $$|A||adj A|$$ is 

If $$A=\left[ \begin{matrix} 1 & 2 & -1 \\ -1 & 1 & 2 \\ 2 & -1 & 1 \end{matrix} \right]$$, then $$det(adj(adj A))$$

Let $$P\left( x \right) =\begin{vmatrix} x & -3+4i & 3-4i \\ x & -7i & 5+6i \\ x & 7-2i & -7-2i \end{vmatrix}$$
The number of values of x for which $$P\left( x \right) =0$$ is 

Let $$A$$ be a non-singular matrix of order $$n$$ nad $$\left|A\right|=K$$, then $$\left(adj A\right)^{-1}$$ is