Update app.py

app.py

@@ -108,8 +108,42 @@ When writing SymPy code to verify solutions:
    - For series calculations, print intermediate terms   
 
 6. Matrix Operations and Systems of Equations:
-   - For systems of equations that might be linearly dependent, use row reduction instead of matrix inversion
-   - Here's the template for handling such systems:
+   - Never use symbolic variables as matrix indices:
+     ```python
+     # CORRECT:
+     i, j = 0, 1  # Use integers for indexing
+     M = Matrix([[1, 2], [3, 4]])
+     element = M[i, j]
+     
+     # INCORRECT:
+     x = Symbol('x')
+     element = M[x, 0]  # This will raise an error
+     ```
+
+   - For matrix analysis, always convert equations to Matrix form:
+     ```python
+     # CORRECT:
+     A = Matrix([[1, 2], [3, 4]])
+     eigenvals = A.eigenvals()
+     
+     # For system of equations:
+     x, y = symbols('x y')
+     system = Matrix([[2, 1], [1, -1]])
+     b = Matrix([5, 1])
+     solution = system.solve(b)
+     ```
+     
+   - For matrix operations with variables:
+     ```python
+     # CORRECT:
+     x = Symbol('x')
+     M = Matrix([[x, 1], [2, 3]])
+     result = M * M  # Matrix multiplication
+     
+     # INCORRECT:
+     M[Symbol('i'), Symbol('j')] = x  # Don't use symbolic indices     
+     ```
+   - For systems of equations that might be linearly dependent, use row reduction instead of matrix inversion. Here's the template for handling such systems:
 
 ```python
 from sympy import Matrix, symbols, solve
@@ -170,7 +204,6 @@ if result == "infinite":
         else:
             print(f"{var} = t")  # use different parameter names for multiple free variables
 ```
-
 Always use this template when working with systems of equations to handle potential linear dependence correctly. """
 
 def load_proof_repository():
@@ -733,7 +766,7 @@ def perform_final_verification(revised_solution):
     """
     Perform a final verification of the revised solution for difficulty level 5 problems.
     """
-    verification_prompt = f"""As an expert mathematician, please carefully verify this revised solution to an advanced (graduate-level) mathematics problem.
+    verification_prompt = f"""As an expert mathematician, please carefully verify this revised solution to an advanced mathematics problem.
 
 Revised Solution to Verify:
 {revised_solution}