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|

Answer

To solve the problem, we need to determine the value of the determinant |A + B|, where matrix B is obtained from matrix A by interchanging any two of its rows.

### Step-by-Step Solution:

1. **Understanding the matrices**:
Let matrix A be represented as:
\[ A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]
Now, if we interchange the first and second rows of A to form B, we have:
\[ B = \begin{pmatrix} d & e & f \\ a & b & c \\ g & h & i \end{pmatrix} \]

2. **Finding A + B**:
We add matrices A and B:
\[ A + B = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} + \begin{pmatrix} d & e & f \\ a & b & c \\ g & h & i \end{pmatrix} = \begin{pmatrix} a + d & b + e & c + f \\ d + a & e + b & f + c \\ g + g & h + h & i + i \end{pmatrix} \]
Simplifying this, we get:
\[ A + B = \begin{pmatrix} a + d & b + e & c + f \\ d + a & e + b & f + c \\ 2g & 2h & 2i \end{pmatrix} \]

3. **Identifying identical rows**:
Notice that the first and second rows of \(A + B\) are identical:
\[ \text{Row 1: } (a + d, b + e, c + f) \quad \text{Row 2: } (d + a, e + b, f + c) \]
Since \(d + a = a + d\), \(e + b = b + e\), and \(f + c = c + f\), we can conclude that Row 1 and Row 2 are identical.

4. **Using the property of determinants**:
A fundamental property of determinants states that if two rows (or columns) of a matrix are identical, then the determinant of that matrix is zero. Therefore, we have:
\[ |A + B| = 0 \]

### Final Answer:
Thus, the value of |A + B| is:
\[ \boxed{0} \]