The sum of the products of the elements of any row of a matrix A with the corresponding cofactors of the elements of the same row is always equal to

Options

A. `|A|`
B. `(1)/(2) |A|`
C. 1
D. 0

Answer

To solve the problem, we need to show that the sum of the products of the elements of any row of a matrix \( A \) with the corresponding cofactors of the elements of the same row is equal to the determinant of the matrix \( A \).

### Step-by-Step Solution:

1. **Define the Matrix:**
Let \( A \) be a \( 3 \times 3 \) matrix represented as:
\[ A = \begin{pmatrix} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{pmatrix} \]

2. **Identify the Elements and Cofactors:**
The elements of the first row are \( A_{11}, A_{12}, A_{13} \).
The corresponding cofactors are denoted as \( C_{11}, C_{12}, C_{13} \).

3. **Write the Expression:**
We need to evaluate the expression:
\[ A_{11}C_{11} + A_{12}C_{12} + A_{13}C_{13} \]

4. **Express the Cofactors:**
The cofactor \( C_{ij} \) is defined as:
\[ C_{ij} = (-1)^{i+j} M_{ij} \]
where \( M_{ij} \) is the minor of the element \( A_{ij} \) (the determinant of the matrix obtained by deleting the \( i \)-th row and \( j \)-th column).

5. **Substitute the Cofactors:**
For the first row, we have:
\[ C_{11} = (-1)^{1+1} M_{11} = M_{11}, \quad C_{12} = (-1)^{1+2} M_{12} = -M_{12}, \quad C_{13} = (-1)^{1+3} M_{13} = M_{13} \]
Therefore, substituting these into our expression gives:
\[ A_{11}M_{11} - A_{12}M_{12} + A_{13}M_{13} \]

6. **Relate to the Determinant:**
The expression \( A_{11}C_{11} + A_{12}C_{12} + A_{13}C_{13} \) is equal to the determinant of matrix \( A \):
\[ A_{11}C_{11} + A_{12}C_{12} + A_{13}C_{13} = \text{det}(A) \]

7. **Conclusion:**
Thus, we conclude that the sum of the products of the elements of any row of a matrix \( A \) with the corresponding cofactors of the elements of the same row is always equal to the determinant of the matrix \( A \):
\[ A_{11}C_{11} + A_{12}C_{12} + A_{13}C_{13} = \text{det}(A) \]

### Final Answer:
The sum of the products of the elements of any row of a matrix \( A \) with the corresponding cofactors of the elements of the same row is always equal to \( \text{det}(A) \).