jee-advanced 2016 Q54

jee-advanced · India · paper1 Complex Numbers Arithmetic Roots of Unity and Cyclotomic Expressions

Let $z = \frac{-1 + \sqrt{3}\,i}{2}$, where $i = \sqrt{-1}$, and $r, s \in \{1,2,3\}$. Let $P = \left[\begin{array}{cc} (-z)^r & z^{2s} \\ z^{2s} & z^r \end{array}\right]$ and $I$ be the identity matrix of order 2. Then the total number of ordered pairs $(r,s)$ for which $P^2 = -I$ is

Let $z = \frac{-1 + \sqrt{3}\,i}{2}$, where $i = \sqrt{-1}$, and $r, s \in \{1,2,3\}$. Let $P = \left[\begin{array}{cc} (-z)^r & z^{2s} \\ z^{2s} & z^r \end{array}\right]$ and $I$ be the identity matrix of order 2. Then the total number of ordered pairs $(r,s)$ for which $P^2 = -I$ is

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