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Question: If\(\begin{bmatrix} \alpha & \beta \\ \gamma & - \alpha \end{bmatrix}\)is to be square root of two r...

If[αβγ−α]\begin{bmatrix} \alpha & \beta \\ \gamma & - \alpha \end{bmatrix}is to be square root of two rowed unit matrix, then a, b and g should satisfy the relation

A

1 + a2 + bg = 0

B

1 – a2 – bg = 0

C

1 – a2 + bg = 0

D

a2 + bg = 1

Answer

1 – a2 – bg = 0

Explanation

Solution

[αβγ−α]\begin{bmatrix} \alpha & \beta \\ \gamma & - \alpha \end{bmatrix}= [1001]\sqrt{\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}}

̃ [αβγ−α]2\begin{bmatrix} \alpha & \beta \\ \gamma & - \alpha \end{bmatrix}^{2}= [1001]\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

̃ [α2+βγ00βγ+α2]\begin{bmatrix} \alpha^{2} + \beta\gamma & 0 \\ 0 & \beta\gamma + \alpha^{2} \end{bmatrix}= [1001]\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}

Hence choice (2) is true.