If A is a nonsingular matrix such that `A A^(T)=A^(T)A` and `B=A^(-1) A^(T)`, then matrix B is

Options

A. involuntary
B. orthogonal
C. idempotent
D. none of these

Explanation

To solve the problem step by step, we need to analyze the properties of the matrix \( B \) defined as \( B = A^{-1} A^T \) under the condition that \( A A^T = A^T A \).

### Step-by-Step Solution:

1. **Given Information**:
We know that \( A \) is a nonsingular matrix and satisfies the condition \( A A^T = A^T A \). We also define \( B \) as:
\[ B = A^{-1} A^T \]

2. **Transpose of B**:
We take the transpose of \( B \):
\[ B^T = (A^{-1} A^T)^T \]
Using the property of transposes, we have:
\[ B^T = (A^T)^T (A^{-1})^T = A (A^{-1})^T \]

3. **Finding the Inverse of A**:
We know that the transpose of the inverse of a matrix is the inverse of the transpose:
\[ (A^{-1})^T = (A^T)^{-1} \]
Therefore, we can rewrite \( B^T \):
\[ B^T = A (A^T)^{-1} \]

4. **Multiplying B by B^T**:
Now, we compute \( B B^T \):
\[ B B^T = (A^{-1} A^T)(A (A^T)^{-1}) \]
This simplifies to:
\[ B B^T = A^{-1} (A^T A) (A^T)^{-1} \]

5. **Using the Given Condition**:
From the condition \( A A^T = A^T A \), we can denote \( A^T A \) as \( C \) (a symmetric matrix). Thus:
\[ B B^T = A^{-1} C (A^T)^{-1} \]

6. **Identity Matrix**:
Since \( A A^{-1} = I \) (the identity matrix), we can conclude:
\[ B B^T = I \]

7. **Conclusion**:
Since \( B B^T = I \), this implies that \( B \) is an orthogonal matrix.

### Final Answer:
The matrix \( B \) is orthogonal.