Question

Let $M$ be an $n \times n$ non-zero skew symmetric matrix. Then the matrix $(I_n - M)(I_n + M)^{-1}$ is always
 

• A. Singular
• B. Symmetric
• C. Orthogonal
• D. Idempotent

Hint:

For skew-symmetric $M$, matrices of the form $(I - M)(I + M)^{-1}$ are always orthogonal due to the Cayley transform property.

Answer 1

The Correct Option is C

Step 1: Recall properties of skew-symmetric matrices.
If $M$ is skew-symmetric, then $M^T = -M$.

Step 2: Compute transpose of given matrix.
Let $A = (I - M)(I + M)^{-1}$. Then \[ A^T = [(I + M)^{-1}]^T (I - M)^T = (I + M^T)^{-1}(I - M^T) = (I - M)^{-1}(I + M). \]

Step 3: Show orthogonality.
\[ AA^T = (I - M)(I + M)^{-1}(I - M)^{-1}(I + M) = I. \] Hence $A$ is orthogonal.

Step 4: Conclusion.
\[ \boxed{(I_n - M)(I_n + M)^{-1} \text{ is orthogonal.}} \]