✅ Two-Dimensional Arrays (2D Arrays)

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✅ 1. Definition of Two-Dimensional Array

A Two-Dimensional Array (2D Array) is a collection of elements of the same data type stored in the form of a table (rows and columns).

✅ It is also called a Matrix Array.

📌 Example:
It looks like a grid:

Row\Col   0    1    2
0        10   20   30
1        40   50   60
2        70   80   90

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✅ 2. Declaration of 2D Array

✅ Syntax (C Language)

datatype array_name[rows][columns];

✅ Example:

int A[3][3];

This means:

• 3 rows
• 3 columns
• Total elements = 3 × 3 = 9

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✅ 3. Initialization of 2D Array

✅ Example:

int A[2][3] = {
    {10, 20, 30},
    {40, 50, 60}
};

Another way:

int A[2][3] = {10,20,30,40,50,60};

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✅ 4. Accessing Elements in 2D Array

Elements are accessed using two indexes:

✅ Format:

A[i][j]

Where:

• i = row number
• j = column number

✅ Example:

• A[0][0] = 10
• A[1][2] = 60

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✅ 5. Memory Representation of 2D Array

A 2D array is stored in contiguous memory, but the elements are stored row-wise in C.

✅ (A) Row Major Order (Used in C / C++)

In Row Major Order, all elements of first row are stored first, then second row, and so on.

✅ Example:

A[0][0] A[0][1] A[0][2] A[1][0] A[1][1] A[1][2]

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✅ Address Calculation Formula (Row Major)

[
LOC(A[i][j]) = Base(A) + [(i \times N) + j] \times W
]

Where:

• Base(A) = Base address of array
• N = Number of columns
• W = Size of each element
• i = row index
• j = column index

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✅ 6. Traversing a 2D Array

To traverse a 2D array, we use nested loops.

✅ Example Program (Traversal):

for(i=0; i<rows; i++)
{
    for(j=0; j<cols; j++)
    {
        printf("%d ", A[i][j]);
    }
    printf("\n");
}

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✅ 7. Operations on 2D Arrays

✅ 1. Insertion

In 2D arrays, insertion means assigning a value to any cell:

✅ Example:

A[1][1] = 99;

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✅ 2. Deletion

Deletion is not direct like linked list. We can:

• Replace value by 0 or -1
• Shift elements (complex)

✅ Example:

A[1][1] = 0;

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✅ 3. Searching

Search an element by scanning all values:

✅ Example:

for(i=0;i<rows;i++)
{
    for(j=0;j<cols;j++)
    {
        if(A[i][j] == key)
        {
            //found
        }
    }
}

Time Complexity: ✅ O(rows × cols)

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✅ 8. Applications of 2D Arrays

2D arrays are used in many real-life and programming problems:

✅ Applications:

1. Matrix representation
2. Student marks table
3. Image processing (pixels)
4. Games (chess board, tic-tac-toe)
5. Graph representation (Adjacency Matrix)

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✅ 9. Advantages of 2D Arrays

✅ Benefits:

• Easy representation of matrix data
• Fast access using index
• Easy to use with loops
• Suitable for tables and grids

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✅ 10. Disadvantages of 2D Arrays

❌ Limitations:

• Fixed size (static memory)
• Memory wastage if not fully used
• Insertion and deletion are difficult
• Requires contiguous memory allocation

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✅ Conclusion

A two-dimensional array stores data in row and column form like a table. It is very useful for representing matrix data such as marks, images, and graphs. In C language, 2D arrays are stored in row-major order, and elements are accessed using two indexes A[i][j].

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