If a matrix B is obtained from a square matrix A by interchanging any two of its rows, then what is |A+B| equal to

Options

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

Explanation

To solve the problem, we need to determine the value of the determinant of the matrix \( A + B \), where \( B \) is obtained by interchanging any two rows of the square matrix \( A \).

### Step-by-Step Solution:

1. **Define the Matrix A**:
Let \( A \) be a square matrix of order \( n \). For simplicity, we can represent it as:
\[ A = \begin{pmatrix} a_{11} & a_{12} & \ldots & a_{1n} \\ a_{21} & a_{22} & \ldots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \ldots & a_{nn} \end{pmatrix} \]

2. **Define the Matrix B**:
Matrix \( B \) is obtained by interchanging any two rows of matrix \( A \). For example, if we interchange the first and second rows, we get:
\[ B = \begin{pmatrix} a_{21} & a_{22} & \ldots & a_{2n} \\ a_{11} & a_{12} & \ldots & a_{1n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \ldots & a_{nn} \end{pmatrix} \]

3. **Calculate \( A + B \)**:
Now, we need to find \( A + B \). The addition of matrices is done element-wise:
\[ A + B = \begin{pmatrix} a_{11} + a_{21} & a_{12} + a_{22} & \ldots & a_{1n} + a_{2n} \\ a_{21} + a_{11} & a_{22} + a_{12} & \ldots & a_{2n} + a_{1n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} + a_{n1} & a_{n2} + a_{n2} & \ldots & a_{nn} + a_{nn} \end{pmatrix} \]
Specifically, the first two rows will be:
\[ A + B = \begin{pmatrix} a_{11} + a_{21} & a_{12} + a_{22} & \ldots \\ a_{21} + a_{11} & a_{22} + a_{12} & \ldots \\ \vdots & \vdots & \ddots \end{pmatrix} \]

4. **Identify Identical Rows**:
Notice that the first and second rows of \( A + B \) are identical because:
\[ a_{11} + a_{21} = a_{21} + a_{11} \quad \text{and} \quad a_{12} + a_{22} = a_{22} + a_{12} \]
This means that the first two rows of the matrix \( A + B \) are the same.

5. **Apply the Determinant Property**:
A fundamental property of determinants states that if a matrix has two identical rows (or columns), then its determinant is zero. Therefore:
\[ |A + B| = 0 \]

### Conclusion:
Thus, the determinant \( |A + B| \) is equal to \( 0 \).