If ${\left\{ {\left( \begin{gathered}
  3\,\,1\,\,2 \hfill \\
  8\,\,9\,\,5 \hfill \\
  1\,\,\,1\,\,3 \hfill \\ 
\end{gathered}  \right)\,\left( \begin{gathered}
  1\,\,3\,\,3 \hfill \\
  3\,\,2\,\,7 \hfill \\
  3\,\,7\,\,9 \hfill \\ 
\end{gathered}  \right)\left( \begin{gathered}
  3\,\,8\,\,1 \hfill \\
  1\,\,\,9\,\,1 \hfill \\
  2\,\,5\,\,3 \hfill \\ 
\end{gathered}  \right)} \right\}^2}\, = \,\left( \begin{gathered}
  a_1\,\,a_2\,\,a_3 \hfill \\
  b_1\,\,b_2\,\,b_3 \hfill \\
  c_1\,\,c_2\,\,c_3 \hfill \\ 
\end{gathered}  \right)$ 

then the value of $|a_2 – b_1| + |a_3 – c_1| + |b_3 – c_2|$ is

Let $A$ be a symmetric matrix such that $|A|=2$ and $\left[\begin{array}{ll}2 & 1 \\ 3 & \frac{3}{2}\end{array}\right] A =\left[\begin{array}{ll}1 & 2 \\ \alpha & \beta\end{array}\right]$. If the sum of the diagonal elements of $A$ is s, then $\frac{\beta s}{\alpha^2}$ is equal to $……….$.

If $A $ is a skew-symmetric matrix of order $ n$, and  $ C$ is a column matrix of order $n \times 1$, then ${C^T}AC$ is

Given $A$ and $C$ are involutary matrices and $B$ is a non-singular matrix, then $(AB^{-1}C)^{-1}$ is equal to –

Let $\quad P=\left[\begin{array}{cc}\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\end{array}\right], A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$ and $Q=P Q P^{ T }$. If $P ^{ T } Q ^{2007} P =\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]$, then $2 a+b-3 c-4 d$ equal to $……………….$.