Let S be the set which contains all possible values of l, m, n, p, q, r for which A=l2−3p00m2−8qr0n2−15 be a non singular idempotent matrix. Then the sum of all the elements of the set S is _____________.

For idempotent matrix, A2 = A⇒ A−1A2=A−1A              (∵A is nonsingular)or A = IThus, non singular idempotent matrix is always a unit matrix. ∴      l2−3=1    ⇒    l=±2m2−8=1    ⇒    m=±3n2−15=1    ⇒    n=±4and p = q = r = 0Hence, the required sum is 0.