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If a Matrix is Symmetric

Certification: Basic Level Accuracy: 40% Submissions: 5 Points: 5

Write a C# program to determine if a given square matrix is symmetric. A matrix is symmetric if it equals its own transpose (i.e., A = A^T). This means that for each element A[i,j] in the matrix, A[i,j] = A[j,i].

Example 1
    
        
  • Input: matrix = [

    [1, 2, 3],
    
  [2, 5, 6],
    
  [3, 6, 9]
    
]
    • 
        
  • Output: true
    • 
        
  • Explanation: 
            
      
                
    • Step 1: Check if the matrix is square (number of rows equals number of columns).
      • 
                
    • Step 2: For each element at position (i,j) where i is not equal to j, verify if the element at position (j,i) is equal.
      • 
                
    • Step 3: The transpose of the matrix is:
                    [

      [1, 2, 3],
      
  [2, 5, 6],
      
  [3, 6, 9]
      
]
      • 
                
    • Step 4: When the original matrix equals its transpose, the matrix is symmetric.
      • 
            
    
        
    •
Example 2
    
        
  • Input: matrix = [

    [1, 2],
    
  [3, 4]
    
]
    • 
        
  • Output: false
    • 
        
  • Explanation:
            
      
                
    • Step 1: Check if the matrix is square (number of rows equals number of columns).
      • 
                
    • Step 2: For each element at position (i,j) where i is not equal to j, verify if the element at position (j,i) is equal.
      • 
                
    • Step 3: The transpose of the matrix is:
                    [

      [1, 3],
      
  [2, 4]
      
]
      • 
                
    • Step 4: When the original matrix does not equal its transpose, the matrix is not symmetric.
      • 
            
    
        
    •
Constraints
    
        
  • 1 ≤ matrix.length ≤ 100
    • 
        
  • matrix[i].length == matrix.length (the matrix is square)
    • 
        
  • -1000 ≤ matrix[i][j] ≤ 1000
    • 
        
  • Time Complexity: O(n²)
    • 
        
  • Space Complexity: O(1)
    •
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Hint
Solution

Solution Hints

    
        
  • Check if the matrix is square (n×n).
    • 
        
  • For each element at position (i,j), verify if element at position (j,i) is equal.
    • 
        
  • If any pair of elements doesn't match, the matrix is not symmetric.
    • 
        
  • Take advantage of symmetry to reduce comparisons (only need to check one half of the matrix).
    • 
        
  • Implement an efficient algorithm that minimizes redundant checks.
    •

Steps to solve by this approach:

 Step 1: Verify the matrix is square by checking if the number of rows equals the number of columns.
 Step 2: Iterate through the matrix, but only check elements where i > j to avoid redundant comparisons.
 Step 3: For each position (i,j) where i > j, compare matrix[i,j] with matrix[j,i].
 Step 4: If any pair doesn't match, return false immediately.
 Step 5: If all pairs match, return true as the matrix is symmetric.

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