Let A = [`a_(ij)`] be a square matrix of order `3xx 3` and |A|= -7. Find the value of `a_(11) A_(11) +a_(12) A_(12) + a_(13) A_(13) `
where`A_(ij)` is the cofactor of element`a_(ij)`

Answer

To solve the problem, we need to find the value of \( a_{11} A_{11} + a_{12} A_{12} + a_{13} A_{13} \) where \( A_{ij} \) is the cofactor of the element \( a_{ij} \) in the matrix \( A \) and the determinant of \( A \) is given as \( |A| = -7 \).

### Step-by-Step Solution:

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

2. **Using the Determinant Property**:
For any square matrix \( A \), the determinant can also be expressed in terms of its cofactors:
\[ |A| = a_{11} A_{11} + a_{12} A_{12} + a_{13} A_{13} \]
This means that the expression \( a_{11} A_{11} + a_{12} A_{12} + a_{13} A_{13} \) is equal to the determinant of the matrix \( A \).

3. **Substituting the Given Determinant**:
Since we know that \( |A| = -7 \), we can substitute this value into the equation:
\[ a_{11} A_{11} + a_{12} A_{12} + a_{13} A_{13} = -7 \]

4. **Final Result**:
Therefore, the value of \( a_{11} A_{11} + a_{12} A_{12} + a_{13} A_{13} \) is:
\[ \boxed{-7} \]