Let `A=[a_(ij)]_(3xx3)` be a matrix such that `a_(ij)=(i+2j)/(2)` where `i,j in [1, 3]` and `i,j inN`. If `C_(ij)` be a cofactor of `a_(ij),` then the value of `a_(11)C_(21)+a_(12)C_(22)+a_(13)C_(23)+a_(21)C_(31)+a_(22)C_(32)+a_(33)C_(33)+a_(31)C_(11)+a_(32)C_(12)+a_(33)C_(13)` is equal to

Answer

To solve the problem, we need to follow these steps:

### Step 1: Construct the Matrix A
Given the formula for the elements of the matrix \( A \) as \( a_{ij} = \frac{i + 2j}{2} \), we will calculate each element of the 3x3 matrix.

- For \( i = 1 \):
- \( a_{11} = \frac{1 + 2 \cdot 1}{2} = \frac{3}{2} \)
- \( a_{12} = \frac{1 + 2 \cdot 2}{2} = \frac{5}{2} \)
- \( a_{13} = \frac{1 + 2 \cdot 3}{2} = \frac{7}{2} \)

- For \( i = 2 \):
- \( a_{21} = \frac{2 + 2 \cdot 1}{2} = 2 \)
- \( a_{22} = \frac{2 + 2 \cdot 2}{2} = 3 \)
- \( a_{23} = \frac{2 + 2 \cdot 3}{2} = 4 \)

- For \( i = 3 \):
- \( a_{31} = \frac{3 + 2 \cdot 1}{2} = \frac{5}{2} \)
- \( a_{32} = \frac{3 + 2 \cdot 2}{2} = \frac{7}{2} \)
- \( a_{33} = \frac{3 + 2 \cdot 3}{2} = 9/2 \)

Thus, the matrix \( A \) is:
\[ A = \begin{bmatrix} \frac{3}{2} & \frac{5}{2} & \frac{7}{2} \\ 2 & 3 & 4 \\ \frac{5}{2} & \frac{7}{2} & \frac{9}{2} \end{bmatrix} \]

### Step 2: Calculate Cofactors
Next, we need to calculate the cofactors \( C_{ij} \) for the matrix \( A \). The cofactor \( C_{ij} \) is given by \( (-1)^{i+j} \cdot M_{ij} \), where \( M_{ij} \) is the determinant of the submatrix obtained by removing the \( i \)-th row and \( j \)-th column.

Calculating the cofactors for the relevant elements:

- \( C_{21} \), \( C_{22} \), \( C_{23} \) (for row 2)
- \( C_{31} \), \( C_{32} \), \( C_{33} \) (for row 3)
- \( C_{11} \), \( C_{12} \), \( C_{13} \) (for row 1)

### Step 3: Apply the Property of Cofactors
From the properties of determinants, we know that the sum of the products of elements of any row with the corresponding cofactors of another row is zero. Thus, we can write:

\[ a_{11}C_{21} + a_{12}C_{22} + a_{13}C_{23} + a_{21}C_{31} + a_{22}C_{32} + a_{23}C_{33} + a_{31}C_{11} + a_{32}C_{12} + a_{33}C_{13} = 0 \]

### Final Step: Conclusion
Since the sum of the products of the elements of any row with the corresponding cofactors of another row is zero, the final value of the expression given in the question is:

\[ \boxed{0} \]