Question

If \[ A= \begin{bmatrix} \cos\alpha& 0 &\sin\alpha\\ 0& 1& 0 -\sin\alpha& 0 &\cos\alpha\\ \end{bmatrix} \] and \(A^2=A^T\) for one value of \(\alpha\in(0,\pi)\), then \(A^3=\):

• A. \(I\)
• B. \(A\)
• C. \(3I\)
• D. \(\begin{bmatrix}0& 0& 00& 3& 00& 0& 0\end{bmatrix}\)

Hint:

Whenever a matrix is orthogonal, replace \(A^T\) by \(A^{-1}\). This usually simplifies the problem immediately.

Answer 1

The Correct Option is A

Concept: The matrix represents rotation about the \(y\)-axis. For an orthogonal matrix, \[ A^T=A^{-1}. \]

Step 1: Use the given condition.
\[ A^2=A^T. \] Since \[ A^T=A^{-1}, \] we get \[ A^2=A^{-1}. \] Multiplying by \(A\), \[ A^3=I. \]

Step 2: Conclude the answer.
\[ \boxed{A^3=I} \] Hence, \[ \boxed{\text{Correct Option (1)}} \]