If A is a skew-symmetric matrix, then trace 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
By definition, for a skew-symmetric matrix $A$, $a_{ij} = -a_{ji}$.
Step 2: Evaluation
For the diagonal elements where $i = j$, we have $a_{ii} = -a_{ii}$, which implies $2a_{ii} = 0$, so $a_{ii} = 0$.
Step 3: Final Calculation
The diagonal elements of any skew-symmetric matrix are always zero.
Step 4: Conclusion
The trace of a matrix is the sum of its diagonal elements. Therefore, Trace of $A = 0 + 0 + ... = 0$.
Hence, the Answer is: (c)

If A is a skew-symmetric matrix, then trace of A is

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The Correct Option is C

Solution and Explanation

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