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The image contains two main problems in linear algebra involving determinants and systems of equations. Here’s an outline of the tasks for each question:

### 1. Determinant Calculation
   The task is to calculate the determinant of the given 4x4 matrix. The matrix is:
   \[
   \begin{vmatrix}
   1 & 7 & n & -2 \\
   3 & \frac{n}{10} & 9 & 8 \\
   5 & -6 & 4 & 8 \\
   n-15 & -2 & 0 & 9 \\
   \end{vmatrix}
   \]
   - **(a)** Expand the determinant along the \(i=3\) row.
   - **(b)** Expand the determinant along the \(j=2\) column.
   - **(c)** Use properties of determinants to simplify and calculate the determinant by bringing it to a diagonal form.

   Here, \( n \) represents the student’s journal number in the group.

### 2. Solving the System of Equations
   The system of equations given is:
   \[
   \begin{cases}
   3x_1 - x_2 + n x_3 = 2 \\
   5x_1 + 2x_2 - 2x_3 = n \\
   \frac{n}{10} x_1 + 4x_2 + (n - 15)x_3 = 2
   \end{cases}
   \]
   This system needs to be solved using four methods:
   - **(a)** Matrix method.
   - **(b)** Cramer’s rule.
   - **(c)** Gaussian elimination.
   - **(d)** Gauss-Jordan elimination.

   Again, \( n \) represents the student’s journal number.

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### Related Questions
1. How do you perform determinant expansion along a specific row or column?
2. What are the properties of determinants that allow simplification to a diagonal form?
3. How is Cramer’s rule applied to a system of equations?
4. What is the difference between Gaussian elimination and Gauss-Jordan elimination?
5. How do you represent a system of equations in matrix form?

#### Tip
When calculating determinants, choosing rows or columns with the most zeros often simplifies calculations.

1. Determinant calculation by expanding along specified row and column, and simplifying using diagonal form.
2. Solving a system of linear equations using matrix, Cramer's rule, Gaussian, and Gauss-Jordan methods.

Explore detailed solutions for determinant calculation and solving a system of linear equations. This problem set covers expansion of determinants, matrix methods, Cramer's rule, Gaussian elimination, and Gauss-Jordan elimination. Suitable for undergraduate students in linear algebra.

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