Determinants Matrices Question 3
Question: If the value of the determinant $ \begin{vmatrix} a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c \\ \end{vmatrix} $ is positive, where $ a\ne b\ne c, $ then the value of abc
Options:
A) Cannot be less than 1
B) Is greater than $ -8 $
C) Is less than $ -8 $
D) Must be greater than 8
Show Answer
Answer:
Correct Answer: B
Solution:
- [b] $ \begin{vmatrix} a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c \\ \end{vmatrix}>0 $
$ \Rightarrow a(bc-1)-1(c-1)+1(1-b)>0 $
$ \Rightarrow abc-a-c+1+1-b>0 $
$ \Rightarrow abc+2-(a+b+c)>0 $
$ \Rightarrow abc>(a+b+c)-2 $ Let; $ a=-1; $
$ b=0 $ & $ c=1 $ Then; $ 0>-2 $ [which is correct] Hence, $ abc=0\Rightarrow abc>-8 $
BETA
