Let S be the set of all possible values of $l,m,n,p,q,r$  for which $A=\left[\begin{array}{ccc}{l}^2-3& p& 0\\ 0& m^2-8& q\\ r& 0& n^2-15\end{array}\right]$  be a non-singular idempotent matrix. Then the sum of all the elements of the set S is………..

For idempotent matrix  $A^2=A$
   $\Rightarrow A^{-1}{A}^2=A^{-1}A$                                         $(\because$ A is nonsingular)
Or  $A=I$
Thus, nonsingular idempotent matrix is always a unit matrix
$\begin{array}{l}\therefore l^2-3=1\Rightarrow l=\pm 2\\ m^2-8=1\Rightarrow m=\pm 3\end{array}$ 
$n^2-15-1\Rightarrow n=\pm 4$  and  $p=q=r=0$
Hence, the required sum is 0.