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.

### Step-by-Step Solution:

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

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

3. **Subtract \(A^2 + B^2\) from both sides:**
\[ 2AB = 0 \]
This implies:
\[ AB = 0 \]

4. **Rearranging gives us:**
\[ AB + BA = 0 \]

5. **Taking the determinant of both sides:**
\[ |AB + BA| = |0| = 0 \]

6. **Using the property of determinants:**
The determinant of a sum is not equal to the sum of the determinants, but we can use the property:
\[ |AB + BA| = |AB| + |BA| \quad \text{(not valid for determinants)} \]
However, since \(AB + BA = 0\), we can analyze the individual terms:
\[ |AB| + |BA| = 0 \]

7. **Knowing that the determinant of a product of matrices is the product of their determinants:**
\[ |AB| = |A||B| \quad \text{and} \quad |BA| = |B||A| \]
Thus:
\[ |A||B| + |B||A| = 0 \]
This simplifies to:
\[ 2|A||B| = 0 \]

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

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