The sum of products of elements of any row with the cofactors of corresponging elements is equal to……………

Explanation

To solve the problem, we need to understand the relationship between the elements of a matrix, their corresponding cofactors, and the determinant of the matrix. We will demonstrate this using a 3x3 matrix as an example.

### Step-by-Step Solution:

1. **Define the Matrix**:
Let's consider a 3x3 matrix \( A \):
\[ A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix} \]

2. **Identify the Row**:
We will consider the first row of the matrix, which consists of the elements \( a_{11}, a_{12}, a_{13} \).

3. **Calculate the Cofactors**:
The cofactor \( C_{ij} \) of an element \( a_{ij} \) is given by \( (-1)^{i+j} \) times the determinant of the submatrix formed by deleting the \( i \)-th row and \( j \)-th column. For the first row, the cofactors are:
- \( C_{11} = \det\begin{pmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{pmatrix} \)
- \( C_{12} = -\det\begin{pmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{pmatrix} \)
- \( C_{13} = \det\begin{pmatrix} a_{21} & a_{22} \\ a_{31} & a_{32} \end{pmatrix} \)

4. **Sum of Products**:
We need to compute the sum of the products of the elements of the first row with their corresponding cofactors:
\[ S = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} \]

5. **Use the Determinant Property**:
According to the properties of determinants, this sum \( S \) is equal to the determinant of the matrix \( A \):
\[ S = \det(A) \]

6. **Conclusion**:
Therefore, the sum of the products of the elements of any row with the cofactors of the corresponding elements is equal to the determinant of the matrix.

### Final Answer:
The sum of products of elements of any row with the cofactors of corresponding elements is equal to the determinant of the matrix.