Let's tackle problem 7(ii) from the assignment.

### Problem 7(ii) Statement:
The problem asks us to prove the identity:

\[
\det(A) = \sum_{j=1}^{n} a_{ij} A_{ij}
\]

where:

- \( A \) is an \( n \times n \) matrix.
- \( a_{ij} \) is the element in the \( i \)-th row and \( j \)-th column of the matrix \( A \).
- \( A_{ij} \) is the cofactor of the element \( a_{ij} \).

### Explanation and Proof:
This statement is similar to the one in 7(i) but involves expanding the determinant along the \( i \)-th row instead of the \( j \)-th column.

#### Determinant Expansion Along the \( i \)-th Row

1. **Cofactor Definition:**

   Just like in 7(i), the cofactor \( A_{ij} \) of the element \( a_{ij} \) is defined as:

   \[
   A_{ij} = (-1)^{i+j} M_{ij}
   \]

   where \( M_{ij} \) is the determinant of the \((n-1) \times (n-1)\) submatrix obtained by removing the \( i \)-th row and \( j \)-th column from \( A \).

2. **Cofactor Expansion Along the \( i \)-th Row:**

   The determinant of the matrix \( A \) can also be expanded along any row, in this case, the \( i \)-th row:

   \[
   \det(A) = \sum_{j=1}^{n} a_{ij} A_{ij}
   \]

   This formula expresses the determinant as the sum of products of the elements in the \( i \)-th row and their corresponding cofactors.

3. **Proof Outline:**

   - The cofactor expansion along a row works in the same way as along a column. By the definition of the determinant, each element of the \( i \)-th row is multiplied by the determinant of the submatrix obtained by removing the corresponding row and column, and the result is alternately added and subtracted (based on the sign given by \( (-1)^{i+j} \)).
   - Summing these products gives us the value of the determinant of the matrix \( A \).

### Conclusion:
This identity \( \det(A) = \sum_{j=1}^{n} a_{ij} A_{ij} \) is a standard result from linear algebra, specifically from the cofactor expansion method, and confirms that the determinant of \( A \) can be computed by expanding along any row, just as it can be expanded along any column.

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Would you like any more details or have any further questions about this proof?

### Related Questions:
1. How does the expansion of the determinant along a row differ from the expansion along a column?
2. Can this method of cofactor expansion be generalized to non-square matrices?
3. How does the concept of cofactor relate to the concept of minors in a matrix?
4. Can we use this identity to derive other properties of determinants, such as the effect of row swaps?
5. How does the cofactor expansion help in understanding the inverse of a matrix?

### Tip:
When expanding a determinant using cofactor expansion, always check if any row or column has zeros, as it can significantly simplify the calculation.

Learn how to prove the identity \( \det(A) = \sum_{j=1}^{n} a_{ij} A_{ij} \) using cofactor expansion. This problem involves linear algebra concepts, including determinants and cofactor expansions, suitable for undergraduate-level studies.

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