Question

If $A = \left(\begin{array}{ccc} 2 & 0 & 0 \\ 0 & \cos x & \sin x \\ 0 & -\sin x & \cos x \end{array}\right)$, then $\text{Adj}(A)^{-1}$

• A. A
• B. 2A
• C. \( \frac{1}{2} A \)
• D. None of these

Hint:

For rotation matrices, the adjugate is often related to the original matrix, and the inverse of the adjugate matrix can be simplified.

Answer 1

The Correct Option is C

Step 1: Understanding Adjoint of a Matrix
The adjugate (or adjoint) of a matrix is the transpose of the cofactor matrix.
For a 3x3 matrix \( A \), the inverse is related to the adjugate by the formula: \[ A^{-1} = \frac{1}{\det(A)} \cdot \text{Adj}(A) \]
Step 2: Finding the Determinant of \( A \)
Since \( A \) is a rotation matrix, its determinant is 1.
Step 3: Conclusion
Thus, \( \text{Adj}(A)^{-1} = \frac{1}{2} A \).