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Mathematics Question on Transpose of a Matrix

If [αβ γ−α]\begin{bmatrix}\alpha&\beta\\\ \gamma&-\alpha\end{bmatrix} is square root of identity matrix of order 22 then

A

1+α2+βγ=0 1 + \alpha^2 + \beta \gamma = 0

B

1+α2−βγ=0 1 + \alpha^2 - \beta \gamma = 0

C

1−α2+βγ=0 1 - \alpha^2 + \beta \gamma = 0

D

α2+βγ=1\alpha^2 + \beta \gamma = 1

Answer

α2+βγ=1\alpha^2 + \beta \gamma = 1

Explanation

Solution

[αβ γ−α]=I2;\begin{bmatrix}\alpha&\beta\\\ \gamma&-\alpha\end{bmatrix} = \sqrt{I_{2}} ;
[αβ γ−α][αβ γ−α]=[10 01]\begin{bmatrix}\alpha&\beta\\\ \gamma&-\alpha\end{bmatrix} \begin{bmatrix}\alpha &\beta \\\ \gamma &-\alpha \end{bmatrix} = \begin{bmatrix}1&0\\\ 0&1\end{bmatrix}
⇒α2+βγ=1\Rightarrow \alpha^{2} + \beta\gamma = 1