If $\omega$ is a root of the equation $x+\frac{1}{x}+1=0$,then the value of the determinant $\left|\begin{array}{ccc}1 & 1+\omega & 1+\omega+\omega^2 \\ 3 & 4+3 \omega & 5+4 \omega+3 \omega^2 \\ 6 & 9+6 \omega & 11+9 \omega+6 \omega^2\end{array}\right|$ is equal to

Let $A$ denote the matrix $\left[\begin{array}{ll}0 & i \\ i & 0\end{array}\right]$,where $i^2=-1$,and let $I$ denote the identity matrix $\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$. Then,$I+A+A^2+\ldots+A^{2010}$ is

If $A = \begin{bmatrix} 3 & 7 \\ 1 & 2 \end{bmatrix}$,then $|A^{2011} - 5A^{2010}|$ is equal to

Let $A_1, A_2, A_3$ be three $A$.$P$.s with the same common difference $d$ and having their first terms as $A, A+1, A+2$,respectively. Let $a, b, c$ be the $7^{\text{th}}, 9^{\text{th}}, 17^{\text{th}}$ terms of $A_1, A_2, A_3$,respectively,such that $\left|\begin{array}{lll} a & 7 & 1 \\ 2b & 17 & 1 \\ c & 17 & 1\end{array}\right|+70=0$. If $a=29$,then the sum of the first $20$ terms of an $A$.$P$. whose first term is $c-a-b$ and common difference is $\frac{d}{12}$,is equal to $........$.

Let $A$ be a symmetric matrix and $B$ be a skew-symmetric matrix,such that $A - B = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$. Then $|A|$ is:

If $z_1 = 2 + 3 \ i$ and $z_2 = 3 + 2 \ i$,where $i = \sqrt{-1}$,then $\begin{bmatrix} z_1 & z_2 \\ -\bar{z}_2 & \bar{z}_1 \end{bmatrix} \begin{bmatrix} \bar{z}_1 & -z_2 \\ \bar{z}_2 & z_1 \end{bmatrix} =$