Question

Let \( A \) be a \( 3 \times 3 \) real matrix such that \( I_3 + A \) is invertible and let \[ B = (I_3 + A)^{-1}(I_3 - A), \] where \( I_3 \) denotes the \( 3 \times 3 \) identity matrix. Then which one of the following statements is true?

• A. If \( B \) is orthogonal, then \( A \) is invertible
• B. If \( B \) is orthogonal, then all the eigenvalues of \( A \) are real
• C. If \( B \) is skew-symmetric, then \( A \) is orthogonal
• D. If \( B \) is skew-symmetric, then the determinant of \( A \) equals \( -1 \)

Hint:

For matrix problems involving orthogonality or skew-symmetry, always check the transpose relationships and use the properties of matrix inverses.

Answer 1

The Correct Option is C

We are given that \( B = (I_3 + A)^{-1}(I_3 - A) \) and need to determine the correct statement.
Step 1: Analyze the properties of \( B \).
We need to find the condition under which \( B \) is skew-symmetric, i.e., \( B^T = -B \). From the definition of \( B \), we have: \[ B^T = \left( (I_3 + A)^{-1}(I_3 - A) \right)^T = (I_3 - A)^T (I_3 + A)^{-1}. \] Since \( A \) is a real matrix, we get: \[ B^T = (I_3 - A) (I_3 + A)^{-1}. \] For \( B^T = -B \), we obtain the condition that \( A \) must be orthogonal, as this ensures that \( (I_3 + A)^{-1}(I_3 - A) \) satisfies the skew-symmetry property. Therefore, the correct answer is (C). Final Answer: \[ \boxed{C}. \]