Question

If \( A \) is an orthogonal matrix, then \( A^{-1} \) is

• A. Symmetric
• B. Skew-symmetric
• C. Orthogonal
• D. Hermitian

Hint:

Orthogonal matrices preserve length and angle, making them useful in rotations and reflections in geometry and linear algebra.

Answer 1

The Correct Option is C

We are given that \( A \) is an orthogonal matrix. By definition, a matrix \( A \) is orthogonal if:

\[ A^T A = A A^T = I \]

Where \( I \) is the identity matrix. The inverse of an orthogonal matrix is also orthogonal. Specifically, for an orthogonal matrix \( A \), we have:

\[ A^{-1} = A^T \]

This means the inverse of an orthogonal matrix is also orthogonal, as its transpose is its inverse.

Thus, the correct answer is option (C), Orthogonal.