If `(A+B)^(2)=A^(2)+B^(2)` and `|A| ne 0` , then `|B|=` (where `A` and `B` are matrices of odd order)

Options

A. `2`
B. `-2`
C. `1`
D. `0`

Explanation

To solve the problem, we need to analyze the equation given and derive the determinant of matrix \( B \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ (A + B)^2 = A^2 + B^2 \]

2. **Expand the left-hand side:**
\[ (A + B)(A + B) = A^2 + AB + BA + B^2 \]
Thus, we can rewrite the equation as:
\[ A^2 + AB + BA + B^2 = A^2 + B^2 \]

3. **Cancel \( A^2 \) and \( B^2 \) from both sides:**
\[ AB + BA = 0 \]

4. **Rearrange the equation:**
\[ AB = -BA \]

5. **Take the determinant of both sides:**
\[ |AB| = |-BA| \]

6. **Use the property of determinants:**
\[ |AB| = |A||B| \quad \text{and} \quad |-BA| = (-1)^n |B||A| \]
Here, \( n \) is the order of the matrices, which is odd.

7. **Substituting the properties into the equation:**
\[ |A||B| = (-1)^n |B||A| \]

8. **Since \( n \) is odd, \( (-1)^n = -1 \):**
\[ |A||B| = -|B||A| \]

9. **Rearranging gives:**
\[ |A||B| + |B||A| = 0 \]
This simplifies to:
\[ |B|(|A| + |A|) = 0 \]
or
\[ |B| \cdot 2|A| = 0 \]

10. **Since \( |A| \neq 0 \) (given in the problem), we can conclude:**
\[ |B| = 0 \]

### Final Answer:
\[ |B| = 0 \]