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 \( B = (I - A)(I + A)^{-1} \) where \( A \) is a skew-symmetric matrix. A skew-symmetric matrix satisfies the property \( A^T = -A \). We will show that \( B \) is an orthogonal matrix.

### Step 1: Properties of Skew-Symmetric Matrices
Given that \( A \) is a skew-symmetric matrix, we have:
\[ A^T = -A \]

### Step 2: Calculate \( B^T \)
To find out if \( B \) is orthogonal, we need to compute \( B^T \):
\[ B = (I - A)(I + A)^{-1} \]
Using the property of transposes, we have:
\[ B^T = \left((I - A)(I + A)^{-1}\right)^T = \left((I + A)^{-1}\right)^T (I - A)^T \]
Using the property of the inverse of a transpose:
\[ B^T = (I + A^T)^{-1} (I - A^T) \]
Substituting \( A^T = -A \):
\[ B^T = (I - A)^{-1} (I + A) \]

### Step 3: Compute \( B B^T \)
Now we compute \( B B^T \):
\[ B B^T = (I - A)(I + A)^{-1} (I - A)^{-1} (I + A) \]
Using the associative property of matrix multiplication:
\[ B B^T = (I - A)(I - A)^{-1} (I + A)(I + A)^{-1} \]
This simplifies to:
\[ B B^T = I \cdot I = I \]

### Conclusion
Since \( B B^T = I \), we conclude that \( B \) is an orthogonal matrix.

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

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