If `A=[[a_(11), a_(12), a_(13)], [a_(21), a_(22), a_(23)], [a_(31), a_(32), a_(33)]]` and `A_(ij)` denote the cofactor of element `a_(ij)`, then `a_(11)A_(21)+a_(12)A_(22)+a_(13)A_(23)=`

Options

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

Answer

To solve the problem, we need to evaluate the expression \( a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23} \) where \( A_{ij} \) denotes the cofactor of the element \( a_{ij} \) in the matrix \( A \).

### Step-by-Step Solution:

1. **Identify the matrix and its elements**:
\[ A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} \]

2. **Define the cofactor \( A_{ij} \)**:
The cofactor \( A_{ij} \) is defined as:
\[ A_{ij} = (-1)^{i+j} \cdot M_{ij} \]
where \( M_{ij} \) is the determinant of the matrix obtained by deleting the \( i \)-th row and \( j \)-th column from \( A \).

3. **Calculate the cofactors \( A_{21}, A_{22}, A_{23} \)**:
- For \( A_{21} \):
\[ A_{21} = (-1)^{2+1} \cdot M_{21} = -M_{21} = -\begin{vmatrix} a_{12} & a_{13} \\ a_{32} & a_{33} \end{vmatrix} = - (a_{12}a_{33} - a_{13}a_{32}) \]
- For \( A_{22} \):
\[ A_{22} = (-1)^{2+2} \cdot M_{22} = M_{22} = \begin{vmatrix} a_{11} & a_{13} \\ a_{31} & a_{33} \end{vmatrix} = a_{11}a_{33} - a_{13}a_{31} \]
- For \( A_{23} \):
\[ A_{23} = (-1)^{2+3} \cdot M_{23} = -M_{23} = -\begin{vmatrix} a_{11} & a_{12} \\ a_{31} & a_{32} \end{vmatrix} = - (a_{11}a_{32} - a_{12}a_{31}) \]

4. **Substitute the cofactors into the expression**:
Now substitute \( A_{21}, A_{22}, A_{23} \) into the expression:
\[ a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23} \]
becomes:
\[ a_{11}(- (a_{12}a_{33} - a_{13}a_{32})) + a_{12}(a_{11}a_{33} - a_{13}a_{31}) + a_{13}(- (a_{11}a_{32} - a_{12}a_{31})) \]

5. **Simplify the expression**:
Expanding this gives:
\[ -a_{11}a_{12}a_{33} + a_{11}a_{11}a_{13}a_{32} + a_{12}a_{11}a_{33} - a_{12}a_{13}a_{31} - a_{13}a_{11}a_{32} + a_{13}a_{12}a_{31} \]
Rearranging and combining like terms:
\[ (-a_{11}a_{12}a_{33} + a_{12}a_{11}a_{33}) + (a_{11}a_{13}a_{32} - a_{13}a_{11}a_{32}) + (-a_{12}a_{13}a_{31} + a_{13}a_{12}a_{31}) = 0 \]

6. **Conclusion**:
Therefore, the final result is:
\[ a_{11}A_{21} + a_{12}A_{22} + a_{13}A_{23} = 0 \]

### Final Answer:
\[ \boxed{0} \]