If any two adjacent rows or columns of a determinant are interchanged in position, the value of the determinant

Options

A. A. Becomes zero
B. B. Remains the same
C. C. Changes its sign
D. D. Is doubled

Answer

To solve the problem, we need to understand the effect of interchanging two adjacent rows or columns in a determinant.

### Step-by-Step Solution:

1. **Understanding the Determinant**:
Let's consider a 3x3 determinant represented as follows:
\[ D = \begin{vmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{vmatrix} \]

2. **Interchanging Two Rows**:
If we interchange two adjacent rows, for example, the first row and the second row, the determinant becomes:
\[ D' = \begin{vmatrix} a_{21} & a_{22} & a_{23} \\ a_{11} & a_{12} & a_{13} \\ a_{31} & a_{32} & a_{33} \end{vmatrix} \]
According to the properties of determinants, interchanging two rows results in the determinant changing its sign. Therefore, we have:
\[ D' = -D \]

3. **Interchanging Two Columns**:
Similarly, if we interchange two adjacent columns, say the first column and the second column, the determinant becomes:
\[ D'' = \begin{vmatrix} a_{12} & a_{11} & a_{13} \\ a_{22} & a_{21} & a_{23} \\ a_{32} & a_{31} & a_{33} \end{vmatrix} \]
Again, the determinant changes its sign:
\[ D'' = -D \]

4. **Conclusion**:
From the above steps, we can conclude that if any two adjacent rows or columns of a determinant are interchanged, the value of the determinant changes its sign.

Thus, the correct answer is **C**: the value of the determinant changes its sign.