Determinants Matrices Question 51
Question: If $ \alpha .\beta .\gamma \in R, $ then the determinant $ \Delta = \begin{vmatrix} {{({e^{i\alpha }}+{e^{-i\alpha }})}^{2}} & {{({e^{i\alpha }}-{e^{-i\alpha }})}^{2}} & 4 \\ {{({e^{i\beta }}+{e^{-i\beta }})}^{2}} & {{({e^{i\beta }}-{e^{-i\beta }})}^{2}} & 4 \\ {{({e^{i\gamma }}+{e^{-i\gamma }})}^{2}} & {{({e^{i\gamma }}-{e^{-i\gamma }})}^{2}} & 4 \\ \end{vmatrix} $ is
Options:
A) Independent of $ \alpha ,\beta $ and $ \gamma $
B) Dependent on $ \alpha ,\beta $ and $ \gamma $
C) Independent of $ \alpha ,\beta $ only
D) Independent of $ \alpha ,\gamma $ only
Show Answer
Answer:
Correct Answer: A
Solution:
- [a] $ C_1\to C_1-C_2 $
$ \Rightarrow \begin{vmatrix} 4 & {{({e^{i\alpha }}-{e^{-i\alpha }})}^{2}} & 4 \\ 4 & {{({e^{i\beta }}-{e^{-i\beta }})}^{2}} & 4 \\ 4 & {{({e^{i\gamma }}-{e^{-i\gamma }})}^{2}} & 4 \\ \end{vmatrix}=0 $
BETA
