Determinants Matrices Question 40
Question: The determinant $ \begin{vmatrix} a+b+c & a+b & a \\ 4a+3b+2c & 3a+2b & 2a \\ 10a+6b+3c & 6a+3b & 3a \\ \end{vmatrix} $ is independent of which one of the following-
Options:
A) a and b
B) b and c
C) a and c
D) All of these
Show Answer
Answer:
Correct Answer: B
Solution:
- [b] $ LetD= \begin{vmatrix} a+b+c & a+b & a \\ 4a+3b+2c & 3a+2b & 2a \\ 10a+6b+3c & 6a+3b & 3a \\ \end{vmatrix} $
$ \Rightarrow D= \begin{vmatrix} a+b+c & a+b & a \\ 4a+3b+2c & 3a+2b & 2a \\ 10a+6b+3c & 6a+3b & 3a \\ \end{vmatrix} $ By $ R_2\to R_2-2R_1 $ and $ R_3\to R_3-3R_1 $ , we get:
$ \Rightarrow \begin{vmatrix} a+b+c & a+b & a \\ 2a+b & a & 0 \\ 7a+3b & 3a & 0 \\ \end{vmatrix} $ By $ C_1\to C_1-C_2 $ gives:
$ \Rightarrow \begin{vmatrix} c & a+b & a \\ a+b & a & 0 \\ 4a+3b & 3a & 0 \\ \end{vmatrix} $ Again by $ R_3\to R_3-3R_1, $ we get: $ D= \begin{vmatrix} a+b+c & a+b & 0 \\ a+b & a & 0 \\ a & 0 & 0 \\ \end{vmatrix} $
$ =a{0.(a+b)-a.a} $
$ =-a^{3} $ which is independent of b and c.
BETA
