Question
If the value of the determinant \[\left| {\begin{array}{*{20}{c}}
a&1&1\\
1&b&1\\
1&1&c
\end{array}} \right|\] is positive, where $$a \ne b \ne c,$$ then the value of $$abc$$
A.
cannot be less than $$1$$
B.
is greater than $$– 8$$
C.
is less than $$– 8$$
D.
must be greater than $$8$$
Answer :
is greater than $$– 8$$
Solution :
\[\left| {\begin{array}{*{20}{c}}
a&1&1\\
1&b&1\\
1&1&c
\end{array}} \right| > 0\]
$$\eqalign{
& \Rightarrow a\left( {bc - 1} \right) - 1\left( {c - 1} \right) + 1\left( {1 - b} \right) > 0 \cr
& \Rightarrow abc - a - c + 1 + 1 - b > 0 \cr
& \Rightarrow abc + 2 - \left( {a + b + c} \right) > 0 \cr
& \Rightarrow abc > \left( {a + b + c} \right) - 2 \cr
& {\text{Let, }}a = - 1;b = 0\,\,\& \,\,c = 1 \cr} $$
Then, $$0 > - 2$$ [which is correct]
Hence, $$abc = 0$$
⇒ $$abc > - 8$$