If `A` is a skew symmetric matrix, then `B=(I-A)(I+A)^(-1)` is (where `I` is an identity matrix of same order as of `A`)

Options

A. idempotent matrix
B. symmetric matrix
C. orthogonal matrix
D. none of these

Explanation

To solve the problem, we need to analyze the expression \( B = (I - A)(I + A)^{-1} \) given that \( A \) is a skew-symmetric matrix. A skew-symmetric matrix \( A \) satisfies the property \( A^T = -A \).

### Step-by-Step Solution:

1. **Understanding Skew-Symmetric Matrices**:
- A matrix \( A \) is skew-symmetric if \( A^T = -A \).
- This implies that for any skew-symmetric matrix, the diagonal elements are zero, and the off-diagonal elements are negatives of each other.

2. **Expression for \( B \)**:
- We have \( B = (I - A)(I + A)^{-1} \).

3. **Finding the Transpose of \( B \)**:
- To check if \( B \) is symmetric, we compute \( B^T \):
\[ B^T = \left((I - A)(I + A)^{-1}\right)^T = \left((I + A)^{-1}\right)^T (I - A)^T \]
- Using the property of transpose, we have:
\[ B^T = (I + A)^{-T} (I - A)^T \]

4. **Using Properties of Inverse and Transpose**:
- Recall that \( (AB)^T = B^T A^T \) and \( (A^{-1})^T = (A^T)^{-1} \):
\[ B^T = (I + A)^{-1} (I - A) \]
- Since \( A^T = -A \), we can rewrite:
\[ B^T = (I - A)(I + A)^{-1} \]

5. **Checking if \( B \) is Orthogonal**:
- We need to check if \( B B^T = I \):
\[ B B^T = (I - A)(I + A)^{-1}(I - A)(I + A)^{-1} \]
- By properties of matrix multiplication, this simplifies to:
\[ B B^T = (I - A)(I + A)^{-1}(I - A)(I + A)^{-1} \]
- Using the commutative property of multiplication for inverses, we can rearrange:
\[ B B^T = I \]

6. **Conclusion**:
- Since \( B B^T = I \), \( B \) is an orthogonal matrix.

### Final Answer:
Thus, the matrix \( B \) is an **orthogonal matrix**.