In C programming, to compute the sum of the main diagonal elements of a square matrix stored as a two dimensional array, which of the following approaches is correct?

Introduction / Context:
This question tests your understanding of how square matrices are represented in C using two dimensional arrays and how to identify the main diagonal elements. The main diagonal of a square matrix consists of elements that share the same row and column index, such as a[0][0], a[1][1], a[2][2] and so on. Knowing how to compute the sum of these elements is a basic matrix operation.

Given Data / Assumptions:

We have a square matrix of size n by n stored as a two dimensional array, for example int a[n][n].
We want the sum of elements on the main diagonal, where row index equals column index.
We assume 0 based indexing, which is standard in C.

Concept / Approach:
The main diagonal of a square matrix is defined as the set of elements a[i][j] such that i equals j. To compute the sum, you can use a single loop that runs from i = 0 to i = n - 1 and add a[i][i] to a running total. There is no need for nested loops if you only care about the main diagonal. Adding the first row, boundaries or sorted middle elements would not generally produce the sum of the diagonal.

Step-by-Step Solution:
Declare an integer variable sum and initialize it to 0. Use a for loop: for (i = 0; i < n; i++). Inside the loop, add the main diagonal element to the sum with sum = sum + a[i][i]. After the loop finishes, sum contains the total of all main diagonal elements a[0][0], a[1][1], ..., a[n - 1][n - 1]. Print or return sum as required by your program.

Verification / Alternative check:
Consider a 3 by 3 matrix: {{1,2,3},{4,5,6},{7,8,9}}. The main diagonal elements are 1, 5 and 9. Using the loop i = 0,1,2 and summing a[i][i] yields 1 + 5 + 9 = 15. Any method that adds other elements such as 2, 4, 6 or 8 will not match the main diagonal sum.

Why Other Options Are Wrong:
Option b adds only the first row, which in the example above is 1 + 2 + 3 = 6, not the diagonal sum of 15.
Option c adds the boundary elements, which include many non diagonal values and ignores inner diagonal elements in larger matrices.
Option d sorting each row and choosing a middle element has no mathematical connection to the main diagonal definition.

Common Pitfalls:
A common mistake is to use a nested loop and test if i equals j inside, which works but is more complicated than a single loop. Another pitfall is to confuse the main diagonal with the secondary diagonal (where i + j equals n - 1), which is a different concept. Always match the indices to the precise diagonal you want.

Final Answer:
To compute the sum of the main diagonal elements, you should loop with i from 0 to n - 1 and accumulate a[i][i] whenever the row and column indices are equal.