If A is an orthogonal matrix, then determinant of A is

Question

If \[ A = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \] and $\theta = \frac{2\pi}{7}$, then $A^{100} = A \times A \times \ldots$ (100 times) is equal to:

Answer 1

Step 1: Understanding the Concept
An orthogonal matrix $A$ satisfies $AA' = I$.
Step 2: Evaluation
Taking the determinant on both sides: $|AA'| = |I|$. Since $|AA'| = |A||A'|$ and $|I| = 1$, we have $|A||A'| = 1$.
Step 3: Final Calculation
Using the property $|A'| = |A|$, the equation becomes $|A|^2 = 1$.
Step 4: Conclusion
Solving for $|A|$, we get $|A| = \pm 1$.
Hence, the Answer is: (c)

If A is an orthogonal matrix, then determinant of A is

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The Correct Option is C

Solution and Explanation

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