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Let P=(a_(i)) be a 5times5 matrix such that a_(ij)=(5)/(2) i=i othwise a_(i)=b 51/2 where b>0.Let alpha beta be smallatt f layeat eyen value of P.9 alpha beta=14.5 then (a) 1 (b) -1 (c) 0 (d) None of these?
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Let P=(a_(i)) be a 5times5 matrix such that a_(ij)=(5)/(2) i=i othwis...


Given Information:
The matrix P is a 5x5 matrix with elements a_(ij) = 5/2 if i=j, otherwise a_(i) = b(5^(1/2)), where b>0. The eigenvalues of P are alpha and beta, and their sum is 14.5.

Solution:

Finding Eigenvalues:
Since alpha and beta are the eigenvalues of P, the sum of the eigenvalues is equal to the trace of the matrix P, which is the sum of the diagonal elements.
Given that a_(ij) = 5/2 if i=j, we have:
a_(11) = a_(22) = a_(33) = a_(44) = a_(55) = 5/2

Therefore, the trace of the matrix P is:
Tr(P) = a_(11) + a_(22) + a_(33) + a_(44) + a_(55) = 5*(5/2) = 12.5

The sum of the eigenvalues alpha and beta is 14.5, so we can write:
alpha + beta = Tr(P) = 12.5

Given that alpha + beta = 14.5, we have:
alpha + beta = 12.5
alpha + beta - 12.5 = 0

Therefore, the correct answer is:
(c) 0
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Let P=(a_(i)) be a 5times5 matrix such that a_(ij)=(5)/(2) i=i othwise a_(i)=b 51/2 where b>0.Let alpha beta be smallatt f layeat eyen value of P.9 alpha beta=14.5 then (a) 1 (b) -1 (c) 0 (d) None of these?
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