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The problem presented is related to **linear programming** and its **dual problem**. Here's the breakdown:

We are given that:
\[
\mathbf{x}^* = (x_1^* = 1, x_2^* = 0, x_3^* = 2)
\]
is an optimal solution to the following linear programming (LP) problem:

### Objective Function (Primal):
\[
\text{Minimize } x_1 + 5x_2 + 2x_3
\]
### Subject to the Constraints:
1. \( x_1 - x_2 \leq 1 \)
2. \( x_1 + x_2 + x_3 \geq 3 \)
3. \( x_1, x_2, x_3 \geq 0 \)

We are asked to evaluate the relationship between \( \lambda_1 \) and \( \lambda_2 \), which represent the optimal solutions of the dual problem corresponding to the constraints of the primal LP.

### Dual Problem:
The dual of the linear programming problem can be formulated by associating dual variables \( \lambda_1 \) and \( \lambda_2 \) with the respective constraints of the primal problem. These variables must satisfy certain relationships because they represent the shadow prices or Lagrange multipliers for the constraints.

From the options, we are tasked with finding which of the relations between \( \lambda_1 \) and \( \lambda_2 \) holds true, given that \( \mathbf{x}^* \) is the optimal solution of the primal and \( (\lambda_1^*, \lambda_2^*) \) is the optimal solution of the dual.

### Key Steps:
1. **Complementary Slackness**: This condition helps relate the primal and dual variables in optimization. Specifically, if a constraint is not tight (not binding), the corresponding dual variable should be zero.
   
2. **Primal-Dual Relationship**: The constraints and objective functions of the primal and dual problems are linked by optimality conditions.

### Answer:
Using these principles, solving the dual relationships should give the correct relationship between \( \lambda_1 \) and \( \lambda_2 \). In this case, after applying complementary slackness and analyzing the constraints, the correct answer is:
\[
\boxed{\text{(A) } 2\lambda_1^* = 3\lambda_2^*}
\]

Would you like to explore the derivation of the dual problem, the process of complementary slackness, or any other details?

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Here are five related questions for further exploration:
1. How is the dual of a linear programming problem formulated?
2. What are the conditions for complementary slackness in linear programming?
3. How do you interpret shadow prices in economic terms from the dual problem?
4. How does the primal solution relate to the feasibility of the dual problem?
5. What methods can be used to solve linear programming problems besides simplex?

**Tip:** Understanding complementary slackness is crucial when solving for optimal solutions in dual problems.

Let (x1* = 1, x2* = 0, x3* = 2) be an optimal solution of the linear programming problem: Minimize x1 + 5x2 + 2x3 subject to x1 - x2 ≤ 1, x1 + x2 + x3 ≥ 3, and x1, x2, x3 ≥ 0. If (λ1*, λ2*) is an optimal solution of its dual, then:

Explore the relationship between the optimal solutions of a linear programming problem and its dual. Given the optimal solution (x1* = 1, x2* = 0, x3* = 2) of the primal problem, determine the correct relationship between dual variables (λ1*, λ2*). This problem involves understanding key concepts such as the primal-dual relationship and complementary slackness.

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