Question

If  \[A = \begin{bmatrix}\alpha & \beta \\ \gamma & - \alpha\end{bmatrix}\]  is such that A² = I, then 

 

Options

• 1 + α² + βγ = 0

• 1 − α² + βγ = 0

• 1 − α² − βγ = 0

• 1 + α² − βγ = 0

   

Answer 1

1 − α² − βγ = 0

\[Here, \]

\[ A^2 = I\]

\[ \Rightarrow \begin{bmatrix}\alpha & \beta \\ \gamma & - \alpha\end{bmatrix}\begin{bmatrix}\alpha & \beta \\ \gamma & - \alpha\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\]

\[ \Rightarrow \begin{bmatrix}\alpha^2 + \beta\gamma & \alpha\beta - \beta\alpha \\ \lambda\alpha - \alpha\gamma & \gamma\beta + \alpha^2\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\]

\[ \Rightarrow \begin{bmatrix}\alpha^2 + \beta\gamma & 0 \\ 0 & \gamma\beta + \alpha^2\end{bmatrix} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\]

\[\]

`"The corresponding elements of two equal matrices are equal ."`

\[ \Rightarrow \alpha^2 + \beta\gamma = 1 \]

\[ \Rightarrow 1 - \alpha^2 - \beta\gamma = 0 \]