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Question: If \(\begin{bmatrix} \alpha & \beta \\ \gamma & - \alpha \end{bmatrix}\)is square root of I<sub>2</s...

If [αβγ−α]\begin{bmatrix} \alpha & \beta \\ \gamma & - \alpha \end{bmatrix}is square root of I2, then a, b and g will satisfy the relation-

A

1 + a2 + bg = 0

B

1 –a2 + bg = 0

C

1 + a2 – bg = 0

D

a2 + bg = 1

Answer

a2 + bg = 1

Explanation

Solution

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

̃ [α2+βγ00α2+βγ]\begin{bmatrix} \alpha^{2} + \beta\gamma & 0 \\ 0 & \alpha^{2} + \beta\gamma \end{bmatrix}= [1001]\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} ̃ a2 + bg = 1