If $\left| {\,\begin{array}{*{20}{c}}{ - {a^2}}&{ab}&{ac}\\{ab}&{ - {b^2}}&{bc}\\{ac}&{bc}&{ - {c^2}}\end{array}\,} \right| = K{a^2}{b^2}{c^2},$ then $K = $
$-4$
$2$
$4$
$8$
For what value of $\lambda $, the system of equations $x + y + z = 6,x + 2y + 3z = 10,$ $x + 2y + \lambda z = 12$ is inconsistent $\lambda =$ ........
Let $A = \left[ {\begin{array}{*{20}{c}}
2&b&1 \\
b&{{b^2} + 1}&b \\
1&b&2
\end{array}} \right]$ where $b > 0$. Then the minimum value of $\frac{{\det \left( A \right)}}{b}$ is
$\left| {\,\begin{array}{*{20}{c}}1&1&1\\a&b&c\\{{a^3}}&{{b^3}}&{{c^3}}\end{array}\,} \right| = $
Let $A=\left(\begin{array}{cc}4 & -2 \\ \alpha & \beta\end{array}\right)$ . If $A ^{2}+\gamma A +18 I = O$, then $\operatorname{det}( A )$ is equal to
If $\left| {\,\begin{array}{*{20}{c}}{x + 1}&1&1\\2&{x + 2}&2\\3&3&{x + 3}\end{array}\,} \right| = 0,$ then $x$ is