Determinants Matrices Question 53
Question: If a, b, c are in GP, then what is the value of $ \begin{vmatrix} a & b & a+b \\ b & c & b+c \\ a+b & b+c & 0 \\ \end{vmatrix}- $
Options:
A) $ 0 $
B) $ 1 $
C) $ -1 $
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
- [a] Since, a, b, c are in GP.
$ \Rightarrow b^{2}=ac $ Expanding the determinant we get, $ \begin{vmatrix} a & b & a+b \\ b & c & b+c \\ a+b & b+c & 0 \\ \end{vmatrix} $
$ =a \begin{vmatrix} c & b+c \\ b+c & 0 \\ \end{vmatrix}-b \begin{vmatrix} b & b+c \\ a+b & 0 \\ \end{vmatrix} $
$ +(a+b) \begin{vmatrix} b & c \\ a+b & b+c \\ \end{vmatrix} $
$ =-a{{(b+c)}^{2}}+b(a+b)(b+c)+(a+b) $
$ (b^{2}+bc-ac-bc) $
$ =-a(b^{2}+c^{2}+2bc)+b(ab+ac+b^{2}+bc) $
$ =-ab^{2}-ac^{2}-2abc+ab^{2}+2abc+b^{2}c $
$ (\because b^{2}=ac) $
$ =-ac^{2}+b^{2}c=-ac^{2}+ac.c=-ac^{2}+ac^{2}=0 $
BETA
