Determinants Matrices Question 149
Question: If $ A= \begin{bmatrix} \alpha & \beta \\ \gamma & \delta \\ \end{bmatrix} $ such that $ A^{2} $ is a two - rowed unit matrix, then $ \delta $ is equal to
Options:
A) $ \alpha $
B) $ \beta $
C) $ \gamma $
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
- [a] We have, $ \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} = \begin{bmatrix} \alpha & \beta \\ \gamma & \delta \\ \end{bmatrix} \begin{bmatrix} \alpha & \beta \\ \gamma & \delta \\ \end{bmatrix} $
$ = \begin{bmatrix} {{\alpha }^{2}}+\beta \gamma & \alpha \beta +\beta \delta \\ \alpha \gamma +\delta \gamma & \beta \gamma +{{\delta }^{2}} \\ \end{bmatrix} $
$ \Rightarrow {{\alpha }^{2}}+\beta \gamma =1,\beta (\alpha +\delta )=0, $
$ \gamma (\alpha +\delta )=0,\beta \gamma +{{\delta }^{2}}=1 $
$ \Rightarrow \beta =0=\gamma ,\alpha \ne -\delta $ and $ {{\alpha }^{2}}={{\delta }^{2}}\Rightarrow \delta =\alpha $
BETA
