Question

If \( A = \frac{1}{\pi} \begin{bmatrix} \sin^{-1} \frac{1}{2} & \tan^{-1} \frac{x}{\pi} \sin^{-1} \frac{x}{\pi} & \cot^{-1} \sqrt{3} \end{bmatrix} \), then \( A - B \) is:

• A. \( 2I \)
• B. \( \frac{1}{2} I \)
• C. \( I \)
• D. 0

Hint:

When subtracting matrices, apply the corresponding inverse trigonometric values and simplify each entry.

Answer 1

The Correct Option is B

Step 1: Simplifying the given matrix.
The matrix \( A \) contains inverse trigonometric functions. Start by simplifying each term in the matrix using known values for \( \sin^{-1} \frac{1}{2} = \frac{\pi}{6} \), \( \cot^{-1} \sqrt{3} = \frac{\pi}{6} \), etc.

Step 2: Finding the value of matrix \( B \).
The matrix \( B \) is similarly constructed and simplifies using standard inverse trigonometric identities. After simplifying, we subtract matrix \( B \) from matrix \( A \).

Step 3: Final result.
After performing the subtraction, we find that \( A - B = \frac{1}{2} I \), which corresponds to option (2).