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Let A=[(l,m,n),(p,q,r),(1,1,1)] and B=A^...

Let `A=[(l,m,n),(p,q,r),(1,1,1)]` and `B=A^(2)`. If `(l-m)^(2)+(p-q)^(2)=9, (m-n)^(2)+(q-r)^(2)=16, (n-l)^(2)+(r-p)^(2)=25`, then the value of `|det(B)-140|` is_______

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Verified by Experts

The correct Answer is:
4
`det(A)` is twice of area of triangle with vertices `(l,p), (m,q),(n,r)` with sides 3, 4,5
`Delta^(2)=s(s-a)(s-b)(s-c)`
`impliesDelta^(2)=6xx3xx2xx1`
`impliesDelta=6`
Now `det(A)=12`
`impliesdet(B)=144`
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