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can we use some identity to find value of ab+by+ya???without finding its eigen values and then multiplying..earlier i have read it somewhere but i forget it.

in Linear Algebra retagged by
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For this we need to find the corresponding characteristic equation which will be cubic :

This is found as :

           | A - λ I |  =  0

==>     (1 - λ)[ (5 -  λ)(1 -  λ) - 1 ]  + 1 [ 3 - 1 + λ ] + 3 [ 1 - 3(5 - λ) ] = 0

==>     λ3  - 7λ2 + 36 = 0   which is the required characteristic equation of the given matrix..

Now using relation between roots and coefficients , we have :

Sum of roots taken two at a time = (Coefficient of λ ) / (Coefficient of λ3)  =   0

Product of roots  =  -(Constant term) /  (Coefficient of λ3)    =  -36

But here it is  -αβγ   = -(-36)  = 36

Hence 36 is the correct answer..

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@habib thanks
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$\lambda$3 - 7$\lambda$2 + 36 = 0

By trial and error, -2 is one of the eigen values.

So, now we can find other eigen values.

So,the equation is:

($\lambda$ + 2)($\lambda$2 -9$\lambda$ + 18) = 0

So, $\lambda$ = -2,6,3

Now, find the value of equation. Enjoy :)