Value of $\left| {\begin{array}{*{20}{c}}
  0&{x - y}&{x - z} \\ 
  {y - x}&0&{y - z} \\ 
  {z - x}&{z - y}&0 
\end{array}} \right|$ is

If $A$, $B$ and $C$ are square matrices of order $3$ such that $A = \left[ {\begin{array}{*{20}{c}}   x&0&1 \\    0&y&0 \\    0&0&z  \end{array}} \right]$ and $\left| B \right| = 36$, $\left| C \right| = 4$,  $\left( {x,y,z \in N} \right)$ and $\left| {ABC} \right| = 1152$ then the minimum value of $x + y + z$ is

If $\left| {\,\begin{array}{*{20}{c}}a&b&{a + b}\\b&c&{b + c}\\{a + b}&{b + c}&0\end{array}\,} \right| = 0$; then $a,b,c$ are in

$\left| {\,\begin{array}{*{20}{c}}{{{\sin }^2}x}&{{{\cos }^2}x}&1\\{{{\cos }^2}x}&{{{\sin }^2}x}&1\\{ - 10}&{12}&2\end{array}\,} \right| = $

If ${a_1},{a_2},{a_3}.....{a_n}....$ are in $G.P.$ then the value of the determinant $\left| {\,\begin{array}{*{20}{c}}{\log {a_n}}&{\log {a_{n + 1}}}&{\log {a_{n + 2}}}\\{\log {a_{n + 3}}}&{\log {a_{n + 4}}}&{\log {a_{n + 5}}}\\{\log {a_{n + 6}}}&{\log {a_{n + 7}}}&{\log {a_{n + 8}}}\end{array}\,} \right|$ is

$\left| {\,\begin{array}{*{20}{c}}{11}&{12}&{13}\\{12}&{13}&{14}\\{13}&{14}&{15}\end{array}\,} \right| = $