Given that `A = [a_(ij)]` is a square matrix of order `3 xx 3 and |A| = - 7`, then the value of `sum_(i=1)^3a_(i2)A_(i2)` where `A_(ij)` denotes the cofactor of element `a_(ij)` is:

Options

A. 7
B. `-7`
C. 0
D. 49

Answer

To solve the problem, we need to find the value of the expression:

\[ \sum_{i=1}^{3} a_{i2} A_{i2} \]

where \( A_{ij} \) denotes the cofactor of the element \( a_{ij} \) in the matrix \( A \).

### Step-by-Step Solution:

1. **Understanding the Cofactor**:
The cofactor \( A_{ij} \) of an element \( a_{ij} \) in a matrix is given by:
\[ A_{ij} = (-1)^{i+j} |M_{ij}| \]
where \( |M_{ij}| \) is the determinant of the matrix obtained by deleting the \( i \)-th row and \( j \)-th column from \( A \).

2. **Applying the Given Information**:
We know that the determinant of \( A \) is given as:
\[ |A| = -7 \]

3. **Using the Property of Determinants**:
There is a property of determinants that states:
\[ \sum_{i=1}^{n} a_{ij} A_{ij} = |A| \]
for any column \( j \) of the matrix \( A \). In our case, we are interested in the second column (i.e., \( j = 2 \)).

4. **Substituting the Values**:
Therefore, we can substitute \( j = 2 \) into the property:
\[ \sum_{i=1}^{3} a_{i2} A_{i2} = |A| = -7 \]

5. **Conclusion**:
Thus, the value of the expression \( \sum_{i=1}^{3} a_{i2} A_{i2} \) is:
\[ -7 \]

### Final Answer:
The value of \( \sum_{i=1}^{3} a_{i2} A_{i2} \) is \( -7 \).