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Community Discussion Question: determinent

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4 May 2008 12:25:16 IST

Subject: determinent

neeraj_agarwal_1990 (909)

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If determinant A= $\begin{vmatrix} a&1 &1 \\ 1&b &1 \\ 1& 1 &c \end{vmatrix}$ is greater than 0, then the value of abc is?

ans-greater than -8

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4 May 2008 13:03:50 IST

Subject: Re:determinent

sboosy (2860)

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$\mbox{Determinant of A} \ = abc-a-b-c+2>0 \ \mbox{(Determinant given to be positive)} \\ \\ abc+2>a+b+c\geq3(abc)^{\frac{1}{3}} \ \mbox{(Second part using AM-GM)} \\ \\ \Rightarrow abc+2>3(abc)^{\frac{1}{3}} \\ \\ \mbox{Let} \ abc=k \ \Rightarrow k^3+6k^2-15k+8>0 \\ \\ \Rightarrow (k-1)^2(k+8) >0 \\ \\ \Rightarrow abc>-8$

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