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

Instructions

The answer to each question is a SINGLE DIGIT INTEGER ranging from 0 to 9, both inclusive.

Question 54

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

Correct Answer: 1

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