The number of positive integral solutions $\left| {\,\,\begin{array}{*{20}{c}}{1 - \lambda }&2&1\\{ - 3}&\lambda &{ - 2}\\2&{ - 2}&{1 + \lambda }\end{array}\,\,} \right|$ $= 0$ is
$0$
$2$
$3$
$1$
Evaluate the determinants
$\left|\begin{array}{rrr}3 & -1 & -2 \\ 0 & 0 & -1 \\ 3 & -5 & 0\end{array}\right|$
If ${\left| {\,\begin{array}{*{20}{c}}4&1\\2&1\end{array}\,} \right|^2} = \left| {\,\begin{array}{*{20}{c}}3&2\\1&x\end{array}\,} \right| - \left| {\,\begin{array}{*{20}{c}}x&3\\{ - 2}&1\end{array}\,} \right|$, then $ x =$
For what value of $\lambda $, the system of equations $x + y + z = 6,x + 2y + 3z = 10,$ $x + 2y + \lambda z = 12$ is inconsistent $\lambda =$ ........
$\left| {\,\begin{array}{*{20}{c}}{13}&{16}&{19}\\{14}&{17}&{20}\\{15}&{18}&{21}\end{array}\,} \right| = $
$\left| {\,\begin{array}{*{20}{c}}1&1&1\\1&{{\omega ^2}}&\omega \\1&\omega &{{\omega ^2}}\end{array}\,} \right| = $