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 ____________

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 _____________.

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Detailed Solution

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.

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