If $AB = A$ and $BA = B$, then

If $n \ne 3k$ and 1, $\omega ,{\omega ^2}$ are the cube roots of unity, then $\Delta = \left| {\,\begin{array}{*{20}{c}}1&{{\omega ^n}}&{{\omega ^{2n}}}\\{{\omega ^{2n}}}&1&{{\omega ^n}}\\{{\omega ^n}}&{{\omega ^{2n}}}&1\end{array}\,} \right|$ has the value

The value of the determinant $\left| {\,\begin{array}{*{20}{c}}2&8&4\\{ - 5}&6&{ - 10}\\1&7&2\end{array}\,} \right|$is

If ${\Delta _r} = \left| {\begin{array}{*{20}{c}}
  r&{2r - 1}&{3r - 2} \\ 
  {\frac{n}{2}}&{n - 1}&a \\ 
  {\frac{1}{2}n\left( {n - 1} \right)}&{{{\left( {n - 1} \right)}^2}}&{\frac{1}{2}\left( {n - 1} \right)\left( {3n - 4} \right)} 
\end{array}} \right|$ then the value of $\sum\limits_{r = 1}^{n - 1} {{\Delta _r}} $

If the system of equations  $2 x+3 y-z=5$  ;  $x+\alpha y+3 z=-4$  ;  $3 x-y+\beta z=7$ has infinitely many solutions, then $13 \alpha \beta$ is equal to

In a third order determinant, each element of the first column consists of sum of two terms, each element of the second column consists of sum of three terms and each element of the third column consists of sum of four terms. Then it can be decomposed into $n $determinants, where $ n$  has the value