If the inverse of $\begin{bmatrix} -x & 14x & 7x \\ 0 & 1 & 0 \\ x & -4x & -2x \end{bmatrix}$ is $\begin{bmatrix} 2 & 0 & 7 \\ 0 & 1 & 0 \\ 1 & -2 & 1 \end{bmatrix}$,then $\left|\begin{array}{ccc} x & x+1 & x+2 \\ x+1 & x+2 & x+3 \\ x+2 & x+3 & x+4 \end{array}\right| = $

Let $A = \begin{bmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{bmatrix}$,$\alpha \in R$ such that $A^{32} = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$. Then a value of $\alpha$ is

$\det \left[ \begin{array}{ccc} \frac{a^2+b^2}{c} & c & c \\ a & \frac{b^2+c^2}{a} & a \\ b & b & \frac{c^2+a^2}{b} \end{array} \right] = $

Let $M$ denote the set of all real matrices of order $3 \times 3$ and let $S=\{-3,-2,-1,1,2\}$. Let $S_1=\{A=[a_{ij}] \in M: A=A^{T} \text{ and } a_{ij} \in S, \forall i, j\}$,$S_2=\{A=[a_{ij}] \in M: A=-A^{T} \text{ and } a_{ij} \in S, \forall i, j\}$,and $S_3=\{A=[a_{ij}] \in M: a_{11}+a_{22}+a_{33}=0 \text{ and } a_{ij} \in S, \forall i, j\}$. If $n(S_1 \cup S_2 \cup S_3)=125 \alpha$,then $\alpha$ equals.

Let $A$ and $B$ be two non-singular skew-symmetric matrices such that $AB = BA$. Then $A^{2} B^{2} (A^{\top} B)^{-1} (A B^{-1})^{\top}$ is equal to

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