Determinants Te...

In a third order determinant $$a_{ij}$$ denotes the element in the ith row and the jth column If $$ a_{ij} = \left\{\begin{matrix}0, & i = j\\ 1, & i > j\\ -1,  & i < j\end{matrix}\right.$$ then the value of the determinant

If $$z = \begin{vmatrix}3 + 3i & 5 - i & 7 - 3i\\ i & 2i & - 3i\\ 3 - 2i & 5 + i & 7 + 3i\end{vmatrix} $$, then

Evaluate $$\begin{vmatrix}cos (x + x^2) & sin(x + x^2) &- cos (x + x^2) \\sin(x - x^2)  & cos (x - x^2) & sin (x - x^2)\\ sin 2x & 0 & sin  2x^2\end{vmatrix}$$

If $$\displaystyle \left | \begin{matrix}a+x &a  &x \\ a-x &a  &x \\ a-x &a  &-x \end{matrix} \right |=0$$ then $$\displaystyle x$$ is

The value of the determinant $$\begin{vmatrix}\sqrt{a} + \sqrt{b} & \sqrt{bc} + \sqrt{2a} & b + \sqrt{ca}\\ 2 \sqrt{c} & c & \sqrt{bc}\\ \sqrt{c} & \sqrt{2c} &c \end{vmatrix}$$ is

The value of the $$\displaystyle m^{th}$$ order determinant of a matrix $$\displaystyle A$$ is $$\displaystyle 15$$ then the value of determinant formed by the cofactors of $$\displaystyle A$$ will be

If the determinant $$\begin{vmatrix}a & b & at-b\\ b & c & bt-c\\ 2 & 1 & 0\end{vmatrix}=0$$, if $$a, b, c$$ are in

The points $$\displaystyle(a, b+c),(b, c+a),(c, a+b)$$ are 

If $$\begin{vmatrix}x^{2}+3x &x+1  &x-2 \\ x-1 &1-2x  &x+4 \\ x+3 &x-4  &3x \end{vmatrix}= Ax^{4}+Bx^{3}+Cx^{2}+Dx+\varrho  $$
Then value of $$\varrho$$ equals to,

The value of the determinant $$\displaystyle \begin{vmatrix}1 & e^{i  \pi/3} &e^{i  \pi/4} \\ e^{- i  \pi/3} &1  &e^{2 i \pi / 3} \\ e^{- i \pi/4} & e^{- 2 i  \pi / 3} & 1\end{vmatrix}$$ is ..................