Let S be the set which contains 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 idem potent matrix. Then, the sum of all the elements of set s=

For idem potent matrix  $A^2=A$
$\Rightarrow A^{-1}{A}^2=A^{-1}A \Rightarrow \left(A^{-1}A\right)A=I$
$A=I$
Thus non-singular idempotent matrix is always a unit matrix.
$\alpha =1^2-3=1 m^2-8=1 n^2-15=1$

$\mathcal{l}=\pm 2 m=\pm 3 n=\pm 4$

$p=q=r=0$

$2-2+3-3+4-4+0+0+0=0$