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Express the relation in $\alpha,\beta,\gamma,\delta \; for \; which \;A\;= \;\left[ \begin{array}{ccc}\alpha+i\gamma & -\beta+i\delta\\ \beta+i\delta & \alpha-i\gamma\end{array} \right] $is Unitary.
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A=$ \left[ \begin{array}{ccc} \alpha + i\gamma & -\beta+i\delta \\ \beta+i\delta & \alpha - i\gamma \end{array} \right] \\ \; \\ \; \\ \therefore \bar{A} \; = \; \left[ \begin{array}{ccc} \alpha - i\gamma & -\beta-i\delta \\ \beta-i\delta & \alpha + i\gamma \end{array} \right] \; and \; A^{\theta} \; = \; \bar{A'} \; = \; \left[ \begin{array}{ccc} \alpha - i\gamma & \beta-i\delta \\ -\beta-i\delta & \alpha + i\gamma \end{array} \right] \\ \; \\ \; \\ A \; square \; matrix \; `A’ \; is \; said \; to \; be \; unitary\; only \; if \; A^{\theta} \cdot A' \; = \; I \\ \; \\ \; \\ \therefore \left[ \begin{array}{ccc} \alpha - i\gamma & \beta-i\delta \\ -\beta-i\delta & \alpha + i\gamma \end{array} \right] \left[ \begin{array}{ccc} \alpha + i\gamma & -\beta+i\delta \\ \beta+i\delta & \alpha - i\gamma \end{array} \right] \; = \; \left[ \begin{array}{ccc} 1 & 0 \\ 0 & 1 \end{array} \right] \\ \; \\ \; \\ \;\\ $ $\left[ \begin{array}{ccc} (\alpha - i\gamma) (\alpha + i\gamma) + (\beta-i\delta)(\beta+i\delta) & (\alpha - i\gamma)(-\beta+i\delta) + (\beta-i\delta)(\alpha - i\gamma) \\ (\alpha + i\gamma)(-\beta-i\delta) + (\beta+i\delta)(\alpha + i\gamma) & (-\beta-i\delta)(-\beta+i\delta) + (\alpha - i\gamma) (\alpha + i\gamma) \end{array} \right] \; = \; \\ \left[ \begin{array}{ccc} 1 & 0 \\ 0 & 1 \end{array} \right] \\ \; \\ \; \\ \; \\ \; \\ \left[ \begin{array}{ccc} \alpha^2-i^2\gamma^2+\beta^2-i^2\delta^2 & (-\alpha\beta+i\alpha\delta+i\beta\gamma-i^2\gamma\delta+\alpha\beta-i\alpha\delta-i\beta\gamma+i^2\gamma\delta) \\ (-\alpha\beta-i\alpha\delta-i\beta\gamma-i^2\gamma\delta+\alpha\beta+i\alpha\delta+i\beta\gamma+i^2\gamma\delta) & \beta^2-i^2\gamma^2+\alpha^2-i^2\delta^2 \end{array} \right] \\ \; \\ \; \\ \; \\ \left[ \begin{array}{ccc} \alpha^2+\beta^2+\gamma^2 +\delta^2 & 0 \\ 0 & \alpha^2+\beta^2+\gamma^2 +\delta^2 \end{array} \right] \; \; \; \ldots \{ \because i^2=\; -1 \} \\ \; \\ \; \\ \; \\ On \; Comparing, \; we \; get \; \alpha^2+\beta^2+\gamma^2 +\delta^2 \; = \; 1 \; \; \; \ldots Ans. $

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